Supervision training supports health care supervisors to perform their essential functions. Realist evaluations are increasingly popular for evaluating complex educational interventions, but no such evaluations exist appraising supervision workshops. Building on an earlier realist synthesis of supervision training, the authors evaluated whether supervision workshops work, for whom and under what circumstances, and why.The authors conducted a 2-stage realist evaluation during 2018–2019 to refine and develop program theory. The intervention involved half-day, face-to-face supervision workshops as part of an Australian state-wide government-funded program for health care and human services supervisors. Data collection involved realist interviews with 10 workshop developers (stage 1) and 43 supervisors (stage 2). The authors employed team-based data analysis using realist logic to refine and develop program theory by identifying contexts, mechanisms, outcomes, and context-mechanism-outcome configurations.Despite their brevity, the supervision workshops had many reported benefits for supervisors  through various perceived mechanisms pertaining to pedagogy , workshops , and individuals . However, they also yielded negative reported outcomes  brought about by assorted perceived mechanisms related to pedagogy , workshops , and individuals . Such mechanisms were thought to be triggered by diverse contexts including supervisors’ levels of experience, sector, and workplace supervision cultures.While the findings partly support the realist synthesis of supervision training and previous realist evaluations of faculty development, this realist evaluation extends this literature considerably. Health care educators should employ mixed pedagogies , relevant content, optimal workshop duration, and competent/engaging facilitators. Educators also need to tailor workshops according to supervisors’ contexts including the sectors and supervision cultures in which supervision is practiced, and supervisors’ levels of experience . Some studies have conducted realist syntheses or evaluations of workplace supervision, 8 but only one study has conducted a realist synthesis of clinical supervision training. 10 Although numerous realist evaluations have been published exploring health education interventions, including coaching skills training, 11 mentoring, 12 leadership development, 13 faculty development, 17 interprofessional learning, 18 knowledge exchange, 19 and preceptorship, 20 none have conducted realist evaluations of supervision training. Therefore, building on a previous realist synthesis of clinical supervision training, 10 we report on a realist evaluation of supervision training workshops to explore whether they work, for whom and under what circumstances, and why.Supervision training for health care and human services professionals matters. Such training can help supervisors enact effectively the 3 essential functions of supervision: helping supervisees learn, supporting supervisees, and ensuring that supervisees perform to standards. 21 largely because this approach focuses on identifying context-triggering mechanisms, which in turn generate outcomes. 25 A glossary of realist terms we employ throughout this article is presented in Supplemental Digital Appendix 1, at http://links.lww.com/ACADMED/B254. 35 Realist evaluations are characterized by 5 concepts:We selected realist methodology for this evaluation, Embeddedness: of human action in social processes or realities;Mechanisms: explanations of how things work;Contexts: the contextual conditions which may enable or hinder mechanisms;Regularities: the patterns of interactions between, for example, contexts and mechanisms; and21Changes: or outcomes generated by context-triggering mechanisms. 21 This methodology is therefore appropriate to guide the testing of a realist synthesis program theory for short-duration supervision training. 10More specifically, realist evaluations seek to illuminate why interventions work or not by building and testing realist program theory, and identifying semipredictable patterns, also known as demi-regularities (DRs) between contexts, mechanisms, and outcomes. These semipredictable patterns are understood in terms of context-mechanism-outcome configurations (CMOCs). 9A previous realist synthesis of the supervision training literature included 29 publications based on 28 studies; with 19 reports focusing on short-duration interventions, defined as one-off interventions or those conducted within 1 week. Pertinent to the current study focusing on one-off workshops, this synthesis identified 10 DRs lettered A–J associated with short-duration interventions. These interventions largely focused on learning outcomes related to supervision knowledge and skills; were delivered face-to-face; and employed mixed pedagogies including didactic, active, and experiential learning. Nine DRs focused on supervisor outcomes (7 with positive and 2 with negative outcomes), whereas the tenth related to supervisee outcomes. Regarding positive supervisor outcomes, DR-A–DR-D related to improved satisfaction, confidence, engagement, knowledge, and practices through the mechanism of mixed pedagogies involving didactic teaching, as well as active and/or experiential learning. DR-E related to improved supervisors’ satisfaction, knowledge, and practices through the mechanism of social relationships. DR-F related to improved supervisory practices through the mechanism of improved supervisor knowledge, skills, and/or attitudes. DR-G identified supervisors’ improved supervision practices through the mechanism of increased supervisor confidence and/or self-efficacy. Finally, the 2 DRs with negative supervisor outcomes were no improvements in supervisory skills through supervisors’ lack of engagement or reinforcement with training (DR-H) and poor supervisor engagement in training through insufficient protected time (DR-I). DR-J related to positive supervisee outcomes, i.e., supervisee development and well-being, through the mechanism of structured training.9 Therefore, the aim of this study’s realist evaluation is to develop this realist program theory by unpacking contexts, to better understand the extent to which a state-wide Australian supervision training program works, for whom and under what circumstances, and why.The program theory based on these 10 DRs for how short-duration supervision training interventions worked and why is depicted in Figure 1 Both workshops covered the different functions of supervision, features of effective supervision, educational principles and practices to support effective supervision, and evaluation of supervisory practices. Delivered by facilitators from varying backgrounds in health care education and supervision, each workshop was limited to 40 participants. Supervisors across the state could register to attend 1 or both of the workshops at their preferred location, free of charge.Funded by the Victoria State Government, we co-designed an evidence-informed, state-wide training program to enhance the supervision knowledge, confidence, and practice of 8,000 health care and human services workers within a funder-determined budget and time frame (2017–2021). Developed by experienced educators, this short-duration intervention—half-day face-to-face workshops—focused on introducing large numbers of supervisors across the state to introductory content; one workshop was directed at supervising students, and another focused on supervising colleagues. The workshops were designed to include mixed pedagogies involving didactic teaching, as well as active reflective and social learning. The workshops explored the diversity of supervision approaches across different disciplines and workplaces, drawing on Proctor’s model of supervision. 9 The aim of stage 1 was to elucidate program developers’ thoughts about how the program might work or not, for whom, and why. In stage 2, we sought to understand supervisors’ perceptions of how the workshops worked or not, under what circumstances, and why. This report follows the RAMESES II reporting standards for realist evaluations. 35We applied a 2-stage evaluation approach to test the realist synthesis program theory. 37 In stage 1, we invited staff from the funder and university who were involved in the design, development, management, and delivery of the supervision workshops to take part in a realist interview. We employed a combination of email and in-person contacts to approach potential participants. In stage 2, we employed email, video, and face-to-face invitations to recruit supervisors attending workshops at different locations. We explicitly encouraged supervisors from diverse professional backgrounds, with different levels of experience, and from varied workplace settings with different supervision cultures to participate. Eligibility criteria were supervisors who attended the workshop(s) and were currently or soon to be supervising students and/or colleagues.In both stages, we used maximum variation sampling to capture diverse perspectives, experiences, and supervisory contexts. 40 The interviewers received feedback from 2 other authors  on their interviewing skills to ensure consistency and adherence to the theory-driven nature of realist methodology. Audio-recorded interviews were conducted face-to-face, by telephone, or via Zoom according to participant convenience.We asked all participants to complete a personal details questionnaire recording demographic and professional characteristics. In both stages, the interviewers  used realist interviewing techniques to elicit understandings about contexts, mechanisms, and outcomes in relation to the workshops. These techniques involved the interviewers seeking information, as well as presenting theories to aid their refinement or development while maintaining a natural flow of conversation. 9 In stage 2, the other interviewers  collected data from the workshop participants following their workshop attendance between May 2019 and September 2019. We aimed to elicit supervisors’ understandings and experiences of the workshops to further refine and develop the realist synthesis program theory. 9 Interview questions with supervisors focused on the contexts in which they undertook their supervision, their reported workshop outcomes, and why they thought the workshops contributed to these outcomes (mechanisms). We conducted 43 interviews , culminating in a total of 35 hours and 47 minutes of data. Note that we also collected longitudinal audio diaries and exit interviews from selected longitudinal cases in 2019 and 2020 to evaluate a different, extended-duration intervention; these data are presented elsewhere. 41In stage 1, a single interviewer (S.L.) interviewed each program developer between August 2018 and December 2018. We asked participants about their expectations of the workshop outcomes, who they thought the workshop would work for or not and why, and the different outcomes that might occur in varying contexts. We conducted 10 interviews, which ranged from 48 to 109 minutes . These interviews provided early information to refine and develop the realist synthesis program theory. 21 to analyze data drawing on the 5 stages used in framework analysis. 42 Throughout these stages, we aimed to refine and develop program theories by identifying contexts, mechanisms, outcomes, and CMOCs.All interviews were transcribed verbatim. We employed a team-based approach using realist logic, While the identification of CMOCs began during data collection, the formal analytic process began post data collection with the whole team familiarizing themselves with 2 diverse subsets of data by independently listening to audio-recordings and reading transcripts: 4 interviews from stage 1 and 8 interviews from stage 2. These subsets were diverse in that they included participants with different involvement in program development/implementation (stage 1) and with varied professional backgrounds, levels of experience, workplace settings, and supervision cultures (stage 2).9 as well as our initial analysis of the subsets of data. The resulting framework added new CMOCs to those already identified from the realist synthesis. 9 A copy of the coding framework is available from the corresponding author (C.E.R.) on request.We developed a comprehensive coding framework partly based on the realist synthesis program theory, 43 We then compared our independent coding of 2 transcripts at the start of the coding process and subsequently verified the coding of 12 other transcripts to ensure coding consistency. Several of us  met monthly to discuss any issues that arose during coding.Two authors  imported all transcripts into NVivo software, version 12 for Windows , and coded contexts, mechanisms, outcomes, and CMOCs line-by-line as per the coding framework. We noted variations to the CMOCs using memos to assist decision making about refining and developing program theory. We used NVivo’s queries function to seek similarities, differences, and patterns in the outcomes and contributing context-triggering mechanisms. We synthesized the coded CMOCs to develop DRs, presenting these alongside causal statements and illustrative quotes. The whole team was included to provide multiple perspectives on the interrogation and theorizing of data.9 to refine and further develop realist program theory.To interpret the data, we compared it with the realist synthesis program theory, 44 Half possessed teaching qualifications (n = 5), the majority had a PhD (n = 6), a health professions education background (n = 6), moderate to expert supervisory expertise (n = 7), and moderate to expert program development expertise (n = 8). Two participants represented themselves as funders, 7 as workshop facilitators, 5 as program evaluators, and 2 as program managers. Note that some had multiple roles.In stage 1, all 10 program developers invited to participate consented. The majority identified as female (n = 8), aged 45–64 years (n = 6), and Oceanian (n = 6), defined as Australian and New Zealand Peoples by the Australian Bureau of Statistics. In stage 2, 51 supervisors expressed interest but only 43 eligible participants consented. These were recruited from 28 workshops in metropolitan (22 workshops) or regional areas (6 workshops). The majority identified as female (n = 34), aged 25–44 years (n = 27), and Oceanian (n = 25). They were mostly from the health care sector (n = 36), and the majority identified as experienced supervisors (n = 28) and described their workplace cultures as supervision-enabled (n = 24), i.e., workplaces with explicit supervision frameworks/arrangements, and organizational supports for supervisors and supervisees . Ten attended the workshop about supervising students only, 18 attended the workshop about supervising colleagues only, and 15 attended both.9 we refined 8 of them into 6 DRs, while finding no evidence for the remaining 2. However, we developed new realist program theory consisting of 6 additional DRs. Given the similarities across the data for stages 1 and 2, as well as our desire to present findings as parsimoniously as possible, we next present a synthesis of the 12 DRs based on the thrust of the mechanisms. Note that these 12 DRs  are a synthesis of 76 CMOCs in our coding framework. As suggested by De Weger and colleagues, 25 we aim to present these rich and complex results as clearly and concisely as possible, and in a way most relevant to, and manageable for, academic medicine professionals. Interested readers can request a copy of our coding framework, including the 76 CMOCs, from the corresponding author (C.E.R.). Importantly, we found evidence of most contexts triggering outcome-generating mechanisms , so below we highlight only the contexts that appeared most likely to trigger outcome-generating mechanisms. These synthesized DRs, with illustrative quotes with contexts [C], mechanisms [M], outcomes [O], and valence  identified, are supplied in Tables  To further facilitate understanding, we visualize these configurations through our modified program theory, which can be seen in Figure In our testing of the 10 DRs identified in the realist synthesis program theory, 9 Firstly, we synthesized these 4 DRs into 1 refined DR based on our realist evaluation . However, we extended the original health care context to include the human services sector, and the mechanism also seemed most likely triggered in non-enabled supervision cultures , workshops , and individuals . Workshop attendees also reported experiencing negative outcomes pertaining to satisfaction, knowledge, and engagement. These negative outcomes were thought to be triggered by mechanisms also related to pedagogy , workshops , and individuals . While these mechanisms were often triggered in assorted contexts including the health care or human services sectors, experienced or inexperienced supervisors, and supervision-enabled or non-enabled cultures, we found some context-specific program theories. Although these findings are partly comparable with the realist synthesis, 46 Furthermore, we did not identify positive supervisor outcomes  found in other faculty development realist evaluations, 17 possibly because our workshop duration was brief. However, we did identify novel positive supervisor outcomes , thereby extending previous realist synthesis findings. 9 Additionally, we identified novel negative workshop outcomes, extending previous faculty development realist evaluations that focused exclusively on positive outcomes only. 17Regarding program outcomes, we found no evidence for supervisee outcomes, probably because we did not explore supervisees’ perceptions as other studies have done. 9 Structured training may not have been found as a mechanism in our evaluation because this was previously found as a mechanism for improved supervisee development and well-being, 46 and we did not identify supervisee outcomes in our evaluation. Furthermore, we did not find confidence/efficacy as a mechanism, probably because this mechanism was found previously for improved supervisory practices for a longer 40-hour intervention. 47 Moreover, our study did not identify positive mechanisms previously identified in faculty development realist evaluations such as professionalization, 17 again probably because our workshops were brief. However, we found a wider variety of mechanisms in our evaluation than previously identified, 9 including those relating to pedagogy , workshop , and individuals . Although some of these have been found in previous realist evaluations of faculty development , 17 these were previously identified as contexts, not mechanisms. Furthermore, we also identified novel negative mechanisms , extending previous faculty development realist evaluations focusing exclusively on positive mechanisms. 17Concerning mechanisms, there were 2 mechanisms identified in the previous realist synthesis for which we found no evidence of . 16 That 1 mechanism, content relevance to supervisor workplace, was most likely triggered for inexperienced supervisors suggests that workplace-related content relevance is especially important for those supervisors. 13 That 2 mechanisms, peer learning and didactic teaching, were most likely triggered in the health care sector, and 2, suboptimal workshop duration and content irrelevance to workplace, in human services, probably reflects differences in supervision cultures, training, and practices across different sectors. 48 For example, in health care contexts, individuals may prefer peer learning and didactic teaching. In human services contexts, individuals may prefer longer training targeted to their sector . That several mechanisms  were most likely triggered in non-enabled supervision cultures highlights the value this brief workshop adds for supervisors with supervisory knowledge or attitudes, who experienced organizational backdrops without established supervision frameworks or managerial support for supervision. Finally, that a single mechanism, didactic teaching, was mostly triggered in supervision-enabled cultures suggests that supervisors with established supervision frameworks or managerial support derive more value from didactic teaching . 49Finally, this evaluation extends the realist synthesis considerably by teasing out contexts that trigger outcome-generating mechanisms. That 2 mechanisms, optimal workshop duration and facilitator competence, were most likely triggered for experienced supervisors suggests that these supervisors may appreciate brief interventions and be more discerning about facilitator competence. Our study is not without its limitations; and these must be considered when thinking through the implications of our findings. For example, the contexts, mechanisms, outcomes, and CMOCs we identified are based on supervisors’ self-reports only, so we cannot be sure whether they concur with objective outcome measures such as supervisor knowledge tests, or supervisees’ perceptions of supervisor outcomes. Furthermore, supervisors reported outcomes for themselves, rather than their supervisees, so we still have limited understanding about the impact of workshops on supervisees. While we employed maximum variation sampling, most participants identified as female and Oceanian and came from the health care sector. Our findings are therefore less transferable to supervisors not identifying as female and those who are culturally and linguistically diverse, as well as those from human services sectors. Finally, although our findings are relatively concordant with the realist synthesis and other realist evaluations of short-duration faculty development programs, they are specific to our half-day supervision workshops designed for health and human services workers.50Despite their brevity, supervision workshops have numerous reported benefits for supervisors through various mechanisms pertaining to pedagogy, workshops, and individuals, as well as within various contexts. Speaking to the mechanisms we identified, medical educators should consider employing several mixed pedagogies including peer learning and didactic teaching, content relevant to supervisors’ workplaces and experience levels, and optimal workshop duration commensurate with experience , and competent and engaging facilitators. However, educators do need to be mindful of contexts triggering those mechanisms, thereby tailoring workshops according to supervisors’ levels of experience, as well as the sectors and supervision cultures in which supervisors supervise. Again, regarding identified mechanisms, for academic medicine supervisors participating in training, we encourage several approaches: be motivated and engaged to embrace the opportunities afforded by workshops, capitalize on their existing knowledge and attitudes within workshops, and protect their time for workshop learning. Finally, for workplace employers, consistent with identified mechanisms, we encourage them to protect academic medicine supervisor time to enable them to engage maximally in supervision workshops. Further research is needed, employing mixed methods, to further refine/develop realist program theory. We encourage realist evaluators to include objective methods such as knowledge tests and subjective approaches with multiple stakeholders such as supervisors and supervisees in future research. We also encourage realist evaluators to include groups other than those that made up our participants, such as males and academic medicine professionals. Finally, this evaluation has focused on supervision training workshops only, and we know from the realist synthesis that intervention duration can lead to different outcomes triggered by different mechanisms within different contexts. Therefore, further realist evaluation is needed to explore extended-duration supervision training and to compare training of different durations. The authors would like to acknowledge colleagues involved in the broader evaluation of the supervision training program: Vicki Edouard, Eve Huang, Charlotte Denniston, Keith Sutton, and Bernadette Ward. They thank all supervision training developers and supervisors who participated in this study for sharing their views and experiences.


Since the problem is, generally speaking, unstable, this requirement is crucial to guarantee that the result following from the explicit algorithm is indeed a solution of the original factorization problem. We prove that a matrix polynomial over the field of Gaussian rational numbers admits the exact Wiener–Hopf factorization if and only if its determinant is exactly factorable. Under such a condition, we adapt the explicit algorithm to the exact calculations and develop the ExactMPF package realized within the Maple Software. The package has been extensively tested. Some examples are presented in the paper, while the listing is provided in the electronic supplementary material. If, however, a matrix polynomial does not admit the exact factorization, we clarify a notion of the numerical (or approximate) factorization that can be constructed by following the explicit factorization algorithm. We highlight possible obstacles on the way and discuss a level of confidence in the final result in the case of an unstable set of partial indices. The full listing of the package ExactMPF is given in the electronic supplementary material.We discuss an explicit algorithm for solving the Wiener–Hopf factorization problem for matrix polynomials. By  By this, we understand that there are explicit formulae allowing us to formally construct the partial indices and the factors of the matrix function Fortunately, there are classes of matrix functions for which an licitly –,13,15. Flicitly – .Unfortunately, the factorization problem for matrix functions is generally speaking unstable. More precisely, if the difference between the larger and the smaller partial indices is greater than one, then the set of the partial indices can change with an arbitrary small perturbation of the initial matrix ,17. Moreexample, , theoremproposed ,–22, undthe key step in construction of any (exact or approximate) factorization of a given matrix function (if the factorization exists) is to provide a procedure/formula allowing for calculation of the partial indices. However, even having an explicit route to achieve the goal, it often requires a numerical realization. And here, one may meet with a hidden unexpected obstacle: the procedure can again be unstable. Below we discuss two simple examples highlighting the issue.Therefore, Consider two matrix polynomials . enough explicit procedure for a particular matrix factorization does not guarantee that it can be successfully used in practice;existence of an  (ii)exact factorization;knowledge of exact values of zeros of the polynomial is not an essential feature (useful though) for execution of the  (iii)explicit, exact and numerical factorizations should be clearly distinguished.the notions of  By the explicit (or constructive) solution of the factorization problem, we understand a clearly defined algorithmic procedure that should definitely terminate after a finite number of steps. However, when one implements a specific explicit factorization algorithm, each consequent step can be executed exactly or approximately .These examples demonstrate thatexplicit algorithm in the exact arithmetic. In such a case, we find that there exists an exact solution of the factorization problem. Unfortunately, as has been discussed above, this resolution is not always available even when in possession of a few different explicit algorithms.To avoid the difficulties associated with possible instability, one can chose the input data belonging to the field the exact solution, the only choice that remains is to construct a numerical (approximate) factorization. The latter, in view of possible instability, may represent a real challenge for execution and all computations should be done with special care and supplemented by accurate analysis of the partial indices preservation and a numerical stability.If a factorization problem does not admit Thus, the key question arising during the factorization of a matrix function is: how to compute the partial indices with a full confidence?There are classes of matrix functions for which the partial indices are known in advance. For example, positive definite matrix functions  have allIn this study, we discuss factorization of matrix polynomials on the unit circle lynomial . We formlynomial .explicit algorithm for factorization of a matrix polynomial, exact arithmetic if and only if its determinant, exact factorization in this field. The solution of the latter is presented in theorem 3.1. We used representation a scalar polynomial as the product of its irreducible factors and the Schur algorithm to test whether each factor has its roots siting inside (or outside) the unit circle. Further, all steps of the explicit matrix factorization can be performed in the exact arithmetic. The algorithm for exact evaluation of the essential polynomials is a foundation stone of theorem 3.2 which is the main result of this section. The formal description of this algorithm is rather complicated and will be presented in the separate work.In §3, we show that all steps of the 1 This package consists of the following basic procedures: SchurTest, SolverExactPF, ExactMLC, seqTk, NumRank, ExactFEP. The main procedure is SolverExactMP. The purposes of these procedures are shortly described in the same section, while the full listing of the package ExactMPF is given in the electronic supplementary material. Several examples of the exact factorizations performed with use of the package ExactMPF are presented in §5 with the aim to demonstrate the package abilities and to study peculiarities of the factorization for special classes of matrix functions. Other tests are used to analyse stability of the computations (see examples 5.1 and 5.2). All those and other examples are included in the electronic supplementary material with more details.In §4, we implemented the exact factorization algorithm as the package ExactMPF in the computer algebra system Maple.In the general case, the explicit algorithm can only be realized approximately. This poses a real challenge for users due to the instability of the problem. Our numerical experiments with ExactMPF have demonstrated that even in the case of unstable sets of partial indices there is a good chance to obtain an approximate factorization. Namely, to accelerate the work of ExactMPF, we provide an option ‘symnum’ into the SolverExactMP. If this option is turn on, the package calculates the ranks of matrices by the singular value decomposition (SVD) method ,27. Thes. 2explicit algorithm for the factorization of matrix polynomials. In accordance with our understanding of the notion of explicit solution (see the Introduction), we describe below those finite numbers of steps that, after their execution, allows one to construct a factorization. We focus on the algorithmic structure of this explicit procedure that should be implemented within a respective numerical code. We would like to emphasize here that this section is not devoted to any study of possible computational problems that may appear during the algorithm implementation into a specific environment.In this section, following the results of the work , we presInput data, that is, a matrix polynomial of a degree invertible on the unit circle  Step 1.Construction of the Wiener–Hopf factorization of the scalar polynomial .We denote It should be noted that any explicit factorization algorithm will use the calculation of the determinant of the factorizable matrix function. The authors realize that the complexity for computing the determinant of a matrix of order Here,  Step 2.Evaluation of a finite set of Laurent coefficients of the matrix function Laurent coefficients of the auxiliary rational matrix function  step in . In whatexplicit algorithm proposed in )Then, equations  imply thAny polynomials As a result, we have defined  (1)(i) the constant terms of the polynomials  (2)(ii) the leading terms of the polynomials By theorem 3.1 from , the seqequation  is regulThe essential polynomials Step 5. Construction of the factorization of the initial matrix polynomial, Letbe the indices andare the right (left) factorization essential polynomials of the sequenceLet us introduce thematrix functionsandThen the left . It is It follows from this theorem that the indices of the sequence ition to , satisfyWe start with explicit algorithm, since their implementation requires only using the linear algebra tools for objects of finite dimensions.The main difficulty in the practical implementation of the aforementioned formulae is that the sequence If, by any means, one believes that the partial indices of a matrix polynomial in question are stable, some computations from Step 4 can be avoided (see example in §6).Those five steps above represent an . 3The analysis of the algorithm, carried out in the previous section, shows the reason for the instability of the factorization problem for matrix polynomials. The explanation is in the instability of the problems of finding the rank and null-space of a matrix. Since finding the rank is the key problem for our algorithm , in theproblems with the factorization instability, with the accuracy of calculation and rounding errors do not arise. It should be noted that symbolic calculations have the significant drawback that they often require significant computational resources. In our case, an increase in the size of the matrix function and the degree of the matrix polynomial can lead to a significant slowdown in the algorithm. Fortunately, in most of the applications, matrix functions of the second or third order mainly appear, and the only reason for a possible slowdown is a large degree of the matrix polynomial. In However, if Steps (1)–(5) can be carried out in exact arithmetic, i.e. the error-free calculations are used, exact factorization of a matrix polynomial explicit algorithm in this particular arithmetic.The main result is formulated in theorem 3.2. It gives a necessary and sufficient condition for the Input data. A matrix polynomial Step 1. In order to construct the Wiener–Hopf factorization of the scalar polynomial exact arithmetic, it is sufficient to require that all roots of However, this condition is not necessary. There exists a weaker necessary and sufficient condition allowing exact Wiener–Hopf factorization of First, we give the criterion in terms of coefficients of the irreducible factors of the polynomial. As has been mentioned above, we assume that no zero lies on the unit circle. Note that, in the opposite case, the Schur test fails during the execution stage. This fact explains why we have not concentrated on how to verify the condition Let For the given Let ynomials :3.1p(z)The polynomialadmits the exact Wiener–Hopf factorization if and only if, for every polynomialone of the matricesoris positive definite.Suppose these conditions are fulfilled. Denote bythe product of those irreducible factorsin decomposition  (ii)the polynomial Since In order to verify that all roots of Step 2. Since the polynomial exact factorization exact computations. Since only a finite number ..deta(z) . 4exact factorization algorithm has been realized in the computer algebra system Maple using its linear algebra utilities as the package ExactMPF. The package can work in the version Maple 11 or higher. Clearly, one can do this in any other suitable software . The main procedure of the package is SolverExactMPF. During the execution, it calls a number of subroutines. We list the basic ExactMPF’s procedures: —SchurTest verifies whether all polynomial roots lie in the unit circle . —SolverExactPF verifies whether a scalar polynomial admits the exact Wiener–Hopf factorization and builds it . —ExactMLC computes the matrix Laurent coefficients equation  and §3,  —seqTk builds the sequence of the matrices  —ExactFEP finds indices and factorization essential polynomials of the sequence The  —are elements of the matrix  —are these polynomials over the field  —does  If the validation fails, the program is interrupted.The main procedure SolverExactMPF finds the exact factorizations of a To access ExactMPF use the commands. > read("ExactMPF.txt"); > with(ExactMPF); > with(LinearAlgebra);To obtain the factorizations of  lplus, dl, lminus, rminus, dr, rplus := SolverExactMPF(a):The procedure SolverExactMPF returns the factors lplus, dl, lminus of the left factorization and the factors rminus, dr, rplus of the right factorization. (i)it finds the sum of the partial indices, (ii)checks whether the factors  (iii)whether This procedure also automatically performs a few tests for the constructed factorizationA very peculiar option symnum is also built into the procedure SolverExactMPF. It can be evaluated by the command: lplus, dl, lminus, rminus, dr, rplus := SolverExactMPF:In this mode, SolverExactMPF converts the matrices NumRank finds a numerical rank of a D method ,27. Recathe spectral condition number of the rectangular matrix Let matrix A ,32.It is known that perturbations of the singular values are of the same order of magnitude as the perturbations of the matrix entries. Thus, computation of the singular values is a stable procedure and this property has led to the definition of a numerical ank see ,27)..τ-rank :‘Failure: The exact polynomial factorization doesn’t exist’‘The factorization process is interrupted’exact factorization of its determinant over the field Consider the second matrix polynomial (z) from  that all > lplus; dl; lminus; > rminus; dr; rplus;Note that the left and right partial indices do not coincide. Moreover, the set of partial indices in the left factorization is stable, while that for the right factorization is unstable. In the next section, we carry out more extensive numerical experiments with ExactMPF.. 5All computations have been performed on a home desktop computer HP with Intel(R) Core(TM)i3-415T CPU, 3.00 GHz, 4G RAM, operating the system Windows 10.explicit factorization of matrix polynomials. In the electronic supplementary material, we include numerous tests performed with use of the ExactMPF package, containing solutions of all examples from the main body of the paper.The package ExactMPF can be used for study of the  (a)First, we consider factorization of triangular matrix functions. In , the rigConsider > rminus, dr, rplus; > lplus, dl, lminus;The execution time is 0.547 s. We note that in this case, the left factorization can be easily constructed without the package, but with another order for the left partial indicesAs is clear from the presented example, the indices of the diagonal elements in the triangular matrix do not always coincide with its partial indices .The factorization of triangular matrix functions of an arbitrary order requires considerations of a number of special cases, depending on the relation between the indices of the diagonal elements . The proLetHere, the indices of diagonal elements are  > lplus, dl, lminus; > drAgain, both the left and right partial indices of this matrix function do not coincide with the indices of the diagonal elements. (b)exact factorization of a matrix polynomial with complex coefficients.In the following example, we construct the  > lplus, dl, lminus, dr;For the same reason as above, we present the factors only for the left factorization. (c)As have been already underlined, the main unsolved problem in the approximate factorization of matrix polynomials is a verification whether its partial exact factorizable matrix polynomials with increasing degrees. We use the full capability of SolverExactMPF in both its modes (exact and symbolic-numeric) to calculate their partial indices and the partial exact arithmetic. If the obtained factors In the next example, we introduce a sequence of A priori, it is clear that the method is ineffective if the task of computation of the The purpose of our experiments is to test applicability of the notion of the As usual, if atlab’s rank command. We can prove thatSince the Taking into account , we see In the next example, we will see how the correctness of such an approach depends on the choice of the parameter digits.The matrix polynomialThe results of the numerical experiment are shown in exact arithmetic, become very slow for large values of The test has demonstrated that the symbolic part of the ExactMPF package is able to deliver the results in reasonable time for exact partial indices would be the same.In the cases indicated by the letter F in le as in . UnfortuThe next example shows that symnum mode is able to correctly calculate the partial indices even in unstable cases.exact computations using SolverExactMPF give Let us consider a matrix polynomialNext, let us construct another matrix polynomial Thus, our numerical experiments with several matrix polynomials  have shown that, for the case of not too large condition numbers, calculations with use of both the symbolic and numeric options of the ExactMPF package give identical partial indices and partial . 6 (i)the input data are given with infinite precision, and (ii)exact (error-free) calculations.there is a possibility to use the  As a result, we have constructed the ExactMPF package for the exact solution of the factorization problem, that returns the solution under condition that the determinate of the matrix polynomial is exactly factorizable or, in the opposite case, interrupts the computations. In this case, the stability and accuracy problems do not arise. However, a natural question arises: can we deliver an approximate  factorization of such a matrix? If the conditions for the existence of the exact factorization are not fulfilled, then it is not yet possible to build any algorithm for the factorization problem that satisfies the entire canon of computational mathematics .In previous sections, we considered the problem of the Wiener–Hopf factorization for matrix polynomial over the field explicit algorithm (Steps 1–5) numerically without relating the steps to the ExactMPF package, in the case, when, at least, one of the conditions (i), (ii) is not fulfilled.Here, we highlight possible obstacles when implementing the Input data. We consider a matrix polynomial Step 1. Since one of the conditions (i), (ii) is not fulfilled, we can solve the factorization problem for the function A standard way for the factorization of a scalar polynomial, recommended by all explicit methods, suggests using approximate values of the polynomial roots. However, those roots are, in general, not well-conditioned functions of polynomial coefficients , and viPolynomialFactorization (in the Maple Software) has been developed by Adukov [guaranteed accuracy In order to solve the factorization problem for scalar polynomials over the field y Adukov . It factPolynomialFactorization package is not included in the ExactMPF package, and is different from the standard ‘Factoring a Polynomial’ Maple command Factor.3 Its main feature is: it does not rely on computation of the polynomial’s roots.Note, that the Step 2. Now we can calculate the Laurent coefficients Hence, we can find the sequence Step 3. This is a crucial point in the factorization problem. In the exact arithmetic, the respective calculations do not meet any problem . However, if the entries of the matrices are known only approximately, evaluations of the ranks and null spaces of NumRank from ExactMPF in detail. It uses the SVD method for finding a numerical rank of matrices with floating point entries. The procedure can also be used in this case.In §4, we have discussed the procedure For definition of the essential polynomials, we need a method for evaluation of a basis of the null space for matrices with floating point entries. This can also be done with the SVD method ,27. CompStep 4. Since entries of the matrices exact. The indices of the sequence indices.Next, to construct an approximate Wiener–Hopf factorization of the matrix polynomial, we need to find the factorization essential polynomial of the sequence Step 5. For the fixed tolerance formulae  with theformulae –2.14) w wStep 5.The key question here remains open:Does the  (i)does a tolerance  (i)can the algorithm for constructing factorization essential polynomials from §3 be adapted to find the factorization essential To solve this problem, we must find answers to the following basic questions:xactMPF package. Since the procedure for finding the factorization essential In order to carry out numerical experiments, we should develop a new appropriate software. At the present time, we have in our hands the package PolynomialFactorization from  and the We consider the matrix polynomial ction in . Note thThus, an approximate factorization  has beenThe Laurent coefficients of formulae  δ0=1, δformulae  gives thIn order to verify the stability criterion for matrix polynomials (theorem 3.1 in ), we neeapleSoftware that returns the singular values of a matrix This can be done with use of the function SingularValues(A) of the MSince the gap between Similarly, the we can expect that the partial indices of Let us represent 3.1 from , the lef. 7explicit algorithm for the exact Wiener–Hopf factorization of a matrix polynomial We have used the explicit algorithm does not necessarily guarantee that the factorization can be effectively performed; and, on the other hand, (b) in contrast to the general opinion, it is indeed possible to find sometimes exact values of the partial indices and then to perform exact factorization, even though the indices are not stable.A message delivered by this paper is twofold: (a) the existence of an exact factorization for matrix polynomials whose coefficients are matrices with entries from the Gaussian rational field exact Wiener–Hopf factorization if and only if the polynomial exactly factorable.One of the main results of this paper is the necessary and sufficient condition for existence of the exactly for a matrix polynomial with the coefficients from the Gaussian rational field The second (and extremely useful) result of the paper is the development of the package ExactMPF in the Maple software that realizes factorization algorithm  —it can be used for symbolic or symbolic-numeral factorization of matrix polynomials. —to analyse stability of the partial indices of matrix polynomials. —to make tests and experiments for verification of proved statements and for a motivation to formulate hypothesis or conjectures for further theoretical development of the theory. They also believe that it may be useful with additional efforts —in some cases to build an approximate factorization of a given matrix polynomial with arbitrary complex coefficients and —to solve real life problems as a computational tool. We shall demonstrate this in the accompanying paper dealing with solving of a discrete analogue of nonlinear Schrödinger equation by the inverse scattering transform method.The use of the ExactMPF package is manyfold. The authors confidently declare that


The QUIC protocol, which was originally proposed by Google, has recently gained a remarkable presence. Although it has been shown to outperform TCP over a wide range of scenarios, there exist some doubts on whether it might be an appropriate transport protocol for IoT. In this paper, we specifically tackle this question, by means of an evaluation carried out over a real platform. In particular, we conduct a thorough characterization of the performance of the MQTT protocol, when used over TCP and QUIC. We deploy a real testbed, using commercial off-the-shelf devices, and we analyze two of the most important key performance indicators for IoT: delay and energy consumption. The results evince that QUIC does not only yield a notable decrease in the delay and its variability, over various wireless technologies and channel conditions, but it does not hinder the energy consumption. In the last few years, we have witnessed a strong increase in Internet of Things (IoT)-based services, having varying characteristics and so posing different requirements to communication protocols. In particular, low latency and energy efficiency are some of the most common challenges of IoT services. Within the IoT field, Industrial IoT (IIoT) is becoming one of the most frequently discussed topics. In fact, it has a notable impact on the economy, since it has the potential to increase industrial productivity by 30% . In 2011IIoT technologies are some of the pillars of the fourth industrial revolution, and they span several fields, including manufacturing, energy, mining, and transportation. In this regard, IIoT can be defined as the combination of smart computing and network technologies applied to industrial processes. This combination seeks the automation of such processes and functionalities within the industry . In thisServices based on IoT, in general, and IIoT, in particular, require the deployment of a large number of devices. On the one hand, these devices need to ensure information and communication reliability. On the other hand, the cost of devices needs to be reduced to afford massive deployments, while access to a power supply cannot be always guaranteed but, in contrast, IIoT devices are frequently battery-driven. Altogether, the need to interconnect a large number of devices with limited hardware capabilities and power resources poses new challenges in terms of energy efficiency, reliability, and security/privacy .In one potential architecture, communication and computation tasks are mostly carried out by devices at the edge of the network, which have constrained resources, including limited battery lifetime. Energy management thus becomes a crucial problem in IIoT . In ordeRecently, the Quick UDP Internet Connections (QUIC) protocol has been developed to overcome some of TCP’s limitations and inefficiencies. This protocol is implemented on top of User Datagram Protocol (UDP) and, among other functionalities, it provides flow and congestion control, reliable communications, in-order delivery to upper layers, and multi-stream management. QUIC was introduced by Google  and stanAll in all, the interplay of IoT protocols, such as MQTT, and QUIC will potentially improve the communication performance of IIoT scenarios. In this sense, this paper introduces a methodology to analyze the performance of MQTT when used over QUIC and TCP over a real platform, using commercial off-the-shelf devices. For this, we design and develop a testbed over which we carry out an evaluation of relevant key performance indicators: delay and energy consumption. This assessment does not only shed light on the suitability of QUIC as a transport protocol for IoT traffic, but it also serves to assess the validity of the proposed methodology.https://github.com/tlmat-unican/MQTT-QUIC-for-RaspberryPi (accessed on 1 April 2022). The implementation includes:Design and implementation of a real testbed based on Raspberry Pi devices. The code developed in the framework is publicly available at –Integration of MQTT over QUIC and TCP in Raspberry Pi devices;–Synchronized setup using Network Time Protocol (NTP) for reliable measurements;–Automation of procedures  for trace generation and result gathering;–traffic control Linux utility; it embraces varying end-to-end delay, bandwidth, and different loss probabilities.Emulation of different channel conditions and technologies, using the Execution of a measurement campaign over such a platform to understand the performance of MQTT over QUIC, and to assess its suitability for IoT services, comparing it with the traditional TCP/Transport Layer Security (TLS) configuration.The aforementioned goal is tackled by means of the following phases, which lead to the main contributions of this paper:The rest of the paper is organized as follows. Since it was initially proposed by Google, the behavior of QUIC has been analyzed under different conditions, but only a few works have focused on studying its performance for IoT services. On the other hand, there do not exist many evaluations of latency and energy consumption when IoT devices use QUIC.A first group of works focuses on establishing the main features of IIoT. In this sense, Mumtaz et al.  analyzedA second group of studies compare the behavior of different application layer protocols, to foster their adoption in IIoT scenarios. Mishra and Kertesz conduct in  a surveyAmong the works devoted to IoT, a number of them focused on the MQTT operation, and in particular on the broker node. For instance, Oliveira et al.  analyze Other works also focus on the MQTT behavior besides the broker node. For instance, Ebleme et al.  assessedAs can be seen, all these works study the performance of application layer protocols for IoT and IIoT. In particular, MQTT stands out as the most widely adopted and analyzed solution. Nevertheless, these IoT studies do not compare the performance of the application layer protocols over different transport solutions, as we do in this work.On the other hand, we can also highlight some works that have analyzed the performance of the QUIC transport protocol. For instance, the authors of  discusseThe authors of  evaluatens-3 [In addition, different methodologies have been exploited to study the performance of QUIC, including simulation and emulation environments. De Biasio et al. described in  a QUIC ins-3  simulatons-3 . The maiOnly a few works have tackled the evaluation of QUIC as a replacement for TCP in IoT or IIoT scenarios. In , HerreroLiri et al.  exploredns-3 network simulator, which is connected to the Linux containers by means of virtual devices , so that the whole protocol stack of the host is not used. The methodology introduced in this work is based on a platform that uses real devices, while underlying channels are emulated at the host itself, instead of exploiting a simulation framework.Two of the closest works to ours are those by Fernández et al. ,46, wherAll in all, the work presented herein complements and broadens the available state-of-the-art, by targeting the evaluation of QUIC as a transport alternative to support IoT protocols, using a real testbed.In this section, we describe the main features of the protocols involved in the evaluation that we discuss later. The aim is to provide a general understanding of their operation, to better tackle the analysis of the performance results in As was already mentioned, MQTT has been widely adopted in the IoT field, in general, and in IIoT, in particular. This protocol usually relies on TCP, which might show shortcomings when used over wireless environments, as well as on TLS, to enable secured end-to-end communications. On the other hand, QUIC, recently standardized by the IETF , may be MQTT is a popular application-level protocol based on the publish–subscribe paradigm , which uThe protocol defines three entities: publisher, subscriber, and broker. Connections are established between publisher and broker, and between subscriber and broker. In all cases, brokers play the server role in traditional client/server architectures, while both publishers and subscribers take the client functionality in their connections.Publishers generate information, whose type is specified by the so-called topic, and send it to the broker. On the other hand, subscribers register at the broker topics they are interested in. Upon receiving messages from publishers, the broker filters them, according to their topic, and forwards them to the interested subscribers. Since MQTT is an application protocol, two independent transport layer connections are established between each pair of entities.QoS 0 (at most once): the message is sent only once, with no retries. Since there is not any acknowledgment at the MQTT level, there is no delivery guarantee per se. Nevertheless, reliable communication can be enforced by lower layers, such as TCP.QoS 1 (at least once): in this case, the message can be retransmitted until the sender receives an acknowledgment. Upon receiving duplicate messages, a flag is set in the corresponding acknowledgment.QoS 2 (exactly once): the message is delivered exactly once. This is ensured with a four-way handshake to ensure that both the original message and its acknowledgment have been correctly received by receiver and sender, respectively. This QoS mode is the slowest one, and it needs, at least, 2 Round Trip Time (RTTs).Another key element that makes MQTT an appropriate application protocol for IIoT services is the support of different QoS levels. There are three supported QoS modes :QoS 0  is the most widespread transport protocol, supporting most of the current Internet services, including those used in IoT environments. It is a connection-oriented solution, offering a reliable service, ensuring in-order delivery, and providing congestion and flow control. In spite of its generalized use, TCP exhibits notable limitations that may affect relevant QoS parameters for IoT. In particular, the congestion control mechanisms usually employed by TCP do not show appropriate performance over wireless links, and they may cause transmission window depletion, thus reducing the transmission rate. Furthermore, TCP single-stream operation could cause HOL blocking, which may lead to longer delays.Moreover, TCP was not originally designed to appropriately handle different traffic types—for instance, IoT and IIoT services—usually characterized by a bursty nature, leading to short connections . TCP reqDue to the aforementioned TCP limitations (among others), many updates or alternatives have emerged, with QUIC  being onQUIC is built on top of UDP, and it includes the TLS 1.3 protocol to guarantee a secure connection. The embedding of TLS within QUIC allows application data to be sent at the first RTT, provided that the endpoints had been previously connected (0-RTT). Opposed to this, and due to the TCP and regular TLS handshake, the start of data transfer in traditional TCP connections can only take place after 2 RTTs, as was mentioned earlier.QUIC information is organized in streams, and it integrates multiplexing techniques to avoid the delay caused by Head-of-Line blocking. This way, when a packet is lost in one stream, only the traffic belonging to this stream is affected. On the other hand, packets in other streams are not hindered, because the orderly reception of streams is not required , thus reIn addition, QUIC brings additional latency reduction, due to its loss detection mechanisms, which include “Early Retransmits” and tail loss probes . These mFurthermore, QUIC features a better upgrading and migration strategy, due to its user-level implementation. Unlike TCP, QUIC is transparent to middleboxes, because it is encapsulated inside the UDP payload. On the other hand, the deployment of TCP enhancements strongly depends on the possibility of updating these middleboxes. While upgrading QUIC might not cause too much trouble, not all devices would be able to simultaneously switch to a newer version. To enable the coexistence of different versions, QUIC includes a version negotiation mechanism, which allows endpoints to decide on the version that they will use for the connection, before it is actually established. This feature allows customization of the protocol with additional features .multi-streaming feature, since we do not evaluate its behavior in the work described herewith.As has been already mentioned, MQTT is one of the most relevant application protocols for IoT and IIoT services. As was stated in Traffic Control (TC) https://tldp.org/HOWTO/html_single/Traffic-Control-HOWTO/ (accessed on 1 April 2022) utilities on the Ethernet interface of the IIoT device. This Linux command allows us to change the interface capacity (bandwidth), to add additional delays, and to establish packet error rates.The IIoT device embeds an MQTT publisher, and it also logs information about traffic generation and energy consumption. The former is obtained using a high-precision multimeter  could be easily used,  such as , where t [TC, or , where N [TC, or ,46,53, tramework , which aramework  link emuThen, the Raspberry Pi devices that implement the broker and subscriber roles are also connected by means of an Ethernet cable. In this case, we do not add any particular characteristic to the link between broker and subscribers in the cloud, since it mimics a wired connection, with a lower impact on the communication. Nevertheless, it is worth noting that additional features could be easily added to this link as well, such as delays over the transport network.golang programming language (quic-go version https://github.com/lucas-clemente/quic-go (accessed on 1 April 2022)).As can be seen in https://github.com/pgOrtiz90/paho.mqtt.golang (accessed on 1 April 2022). As for the broker, we use the Volant MQ https://github.com/VolantMQ/volantmq (accessed on 1 April 2022) implementation (version v0.4.0-rc6). The regular version provides an MQTT (version https://github.com/fatimafp95/volantmq_2 (accessed on 1 April 2022).In addition, we use, for the publisher and subscriber, an MQTT client based on Eclipse-Paho, which can be configured to run over both TCP and the aforementioned QUIC implementation We guarantee the synchronization of devices using NTP, taking the broker internal clock as a reference. It is worth noting that synchronization is performed out-band, and it does not thus affect data traffic. Both the NTP server and client are integrated within the broker and publisher/subscriber devices, respectively. Thanks to this synchronization, we are able to precisely establish the transmission delay between receiver and sender, in order to obtain meaningful statistics, such as the average delay or its coefficient of variation, when using TCP and QUIC.TC configuration to mimic different wireless technologies: WiFi, cellular, and satellite links. In addition, both the QoS used by MQTT, as well as the number of transmitted packets and its sending rate, are also established.The experimentation has been automated by means of Python scripts. In each experiment, these scripts set the pcap files are also generated to better study the actual number of packets that are injected into the interface. In this sense, it is worth mentioning that the MQTT QoS setup, the particular features of TCP and QUIC , and the interplay between application and transport protocols may strongly increase the number of packets that are actually sent in an experiment.As a result, two log files are generated at the publisher and subscriber, comprising time instants of sending and reception events of each packet, as well as the energy consumption in the Raspberry Pi that mimics the IIoT device (publisher). In addition, on the publisher side, This section discusses the performance characterization that was carried out over the testbed previously described. We analyze the behavior of MQTT, over both TCP and QUIC, in terms of traffic delay and energy consumption. In all cases, TCP is used together with TLS, so that all the analyzed configurations provide the same functionalities and can be thus fairly compared.TC utility to mimic the characteristics of different wireless technologies. As mentioned before, we use the QoS 0 level, to keep the communication as simple as possible. For each configuration, we send 1000 packets with a 1 s interval between two consecutive packets. This low rate ensures that we do not congest the channel, and any delay variation is thus caused by the transport protocol behavior. Although analyzing the delay in higher load conditions might also be interesting, other mechanisms may come into play, making the analysis more complex. In particular, under such conditions, the congestion control mechanisms used in TCP and QUIC would modulate the response of the transport protocol. This evaluation is left for our future work.We first measure the delay suffered by the MQTT traffic. We configure MQTT to use the y-axis in each plot corresponds to the link delay of the corresponding technology: 25, 100, and 600 ms for WiFi, cellular, and satellite links, respectively. In general, we can observe that the delay experienced with QUIC is lower than with TCP. In addition, the results evince that QUIC exhibits a rather stable behavior as we increase the loss probability. On the other hand, the delay experienced with TCP increases more abruptly as the loss probability grows.First, in In this sense, the delay when using TCP over WiFi grows up to >40 ms with a loss probability of In the case of QUIC, the results are rather different. When using WiFi, QUIC hardly adds 12 ms of delay for the worst channel conditions, and a similar increase is observed for the cellular technology. As for the satellite link, the delay increase observed for QUIC is ≈35 ms, compared to the Then, in As can be observed in The results in jitter), Similar results are seen over the cellular link; see All in all, the results evince that not only the delay exhibited by TCP is larger than that observed for QUIC, but the variability of such delay is also remarkably lower when using QUIC. In addition, the results show that QUIC’s jitter is more affected by the error rate.We now analyze the impact that the MQTT QoS level has over the delay. In this case, we only show the results obtained with QUIC, since the differences with TCP-based communications are similar to the ones discussed so far. jitter, for the satellite link, when the error rate grows.We complete the delay analysis with vior see . In the Altogether, the analysis shows that the use of higher MQTT QoS levels may have a positive impact when low-latency technologies are used. On the other hand, for technologies with longer RTTs, such as satellite, higher QoS levels actually lead to worse performance as the error rate grows.As was mentioned before, one of the most critical performance indicators to be considered in any evaluation within the context of IIoT is energy consumption. In order to characterize it, we discuss now the results that were obtained for MQTT over QUIC and TCP. We exploited a high-precision digital multimeter , which is defined as the time elapsed between two consecutive packets at the sender. Using the multimeter, we measure the total energy consumed, and we divide it by the time required to send all packets, to obtain comparable measurements of the average power, in Joules per second (J/s). pcap files to count the overall number of packets sent at the physical layer. Although this approach does not allow us to know the actual energy consumed in each measurement, it enables a fair comparison between TCP and QUIC in terms of energy consumption, since it would be proportional to the number of transmissions.Finally, we have broadened the energy consumption analysis by studying the number of actual transmissions when using MQTT over both TCP and QUIC, and for the different wireless technologies that were previously analyzed. In this case, we use an IaT of 1 s, to avoid triggering congestion control mechanisms, as we did in the previous characterization. However, due to the memory limitations of the multimeter, we could not monitor the consumed energy during the whole experiment’s duration. Instead, we use the generated The results show that there might be a small penalization when using QUIC in terms of energy consumption for some of the configurations that we have evaluated (satellite). However, the same conclusion might not be valid for different scenarios—for instance, considering other application protocols . In this sense, it would be necessary to develop more generic energy models—for instance, exploiting complexity-related approaches.In this paper, we have analyzed the performance of one of the most widespread application protocols for IIoT, MQTT, and its interplay with different transport protocols: TCP and QUIC. In this sense, we have seen that, despite its growing relevance, there do not exist many works that have analyzed the performance of QUIC as a transport protocol alternative for IoT traffic. This work sheds light on whether it is a suitable alternative for MQTT-based applications.Furthermore, we introduce an evaluation methodology, which entails the design and development of an evaluation testbed, comprising Raspberry Pi devices, able to provide delay- and energy-related metrics. Over such a framework, we have carried out an extensive measurement campaign to analyze the performance of MQTT/QUIC and MQTT/TLS/TCP stacks over a variety of channel conditions, which emulate different wireless technologies.The obtained results show that the adoption of QUIC as a transport solution may bring relevant benefits in terms of delay over all the analyzed wireless technologies. In this sense, we observed a delay reduction of 25.5% over WiFi links when the loss rate was 5%. For cellular and satellite connections, characterized by longer RTTs, the reduction is less relevant, but QUIC still yields lower delays .More interestingly, the use of QUIC also leads to a sharp jitter (delay variation) mitigation, especially over error-prone links. We studied the RSD of the delay, and the reduction that was observed in our experiments was 61%, 42%, and 41%, for the WiFi, cellular, and satellite links, respectively, when the loss rate was 5%.Focusing on the interplay between MQTT and QUIC, we have observed that the use of MQTT QoS levels has a positive impact in terms of delay over some particular scenarios, especially those having shorter RTTs. On the other hand, the energy measurements obtained with a high-precision multimeter evince that QUIC does not significantly increase the energy consumption. We have actually seen that both TCP and QUIC generate a comparable amount of traffic.Based on the results of the measurement campaign carried out over a platform featuring commercial off-the-shelf devices, we can thus conclude that QUIC is indeed a suitable transport protocol alternative for typical IoT scenarios.NetEm or MahiMahi, to mimic different channel conditions. We will also analyze the impact of using other congestion control solutions, such as BBR, and their suitability for IIoT scenarios. Finally, we will exploit the proposed methodology and the implemented platform to develop more generic energy models, which could be applied to a wider range of configurations.In our future research, we will broaden the work presented here in different ways. First, we will extend the testbed to include other application protocols, such as ZeroMQ, CoAP, or OMA LwM2M. In addition, we will add more complex channel emulators, such as


Drosophila melanogaster Mutations") is a direct descendent of the computer program "Drosophila Viewer" (This is a short notice announcing the publication of an e-book by the Brazilian Society of Genetics (SBG) and reviewing it. The publication , sponsored by the University of Indiana at Bloomington, Indiana. The FlyBase site houses an incredible number of files dedicated to the detailed, in-depth biology of Drosophila, including the excellent collection of photographs of classical phenotypic mutants by  At approximately the same time the program "Drosophila Viewer" was produced, the internet online resources on Drosophila, the e-book was produced by working on all the original drawings used in the program and adding material obtained with permission from other sources, especially from Professors Thomas C. Kaufman , as well as their availability in reference Drosophila research and teaching centers. Each presentation consists of a box containing illustrations and photos of mutants. Each mutant is explained by a short text taken from the public domain Drosophila "red book" by In order to keep alive the graphic material from the "Drosophila Viewer" program and organize everything as an elementary Atlas on phenotypic mutations of Drosophila melanogaster; a few more original color diagrams were produced to represent some new variants from some photographs not included in the computer program or from the photos by Drosophila research group, who described the mutation. The boxes contain also original color photographs of whole-mounts of Drosophila mutants and/or photos of living specimens obtained from the FlyBase repository of Each box contains a diagrammatic but detailed color representation of the corresponding phenotypic mutation and some extra material. The color illustrations, taken from the archives of the program "Drosophila Viewer" had the brown color of the flies reworked to bring it to the true color of the wild-type singed. The box contains, upper row, left: an original color illustration taken and repainted from the "Drosophila Viewer" program, showing the bristles in the mutant and in the wild-type fly; upper row, right: a black & white drawing of a mutant fly  by yellow white female (yw/yw) crosses. The F2 progeny of such crosses is formed by non-recombinant  and by recombinant  flies, due to gene recombination that occurred during gametogenesis in their female parent. The different F2 phenotypes depicted occur with expected frequencies that depend on the physical distance between the involved linked genes. Formal permission for all extra material included in the Atlas was obtained directly from the respective corresponding authors and editors listed in the Atlas and in the program "Drosophila Viewer", or in the publications describing it a,b.


It is essential that nurses quickly learn the proper methods for preventing and controlling nosocomial infection and managing intensive care patients during the COVID-19 pandemic, including the donning and doffing of personal protective equipment (PPE). Virtual reality (VR) simulation offers the advantage of learning in a safe environment with a sense of realism similar to that of an actual clinical setting and has been reported to enhance self-efficacy in infection control, safety performance, and learning satisfaction among students.This study aims to develop a virtual reality infection control simulation (VRICS) program regarding donning and doffing of PPE and respiratory care for pediatric patients admitted to an isolation unit for COVID-19 and to identify the effects of the program on PPE knowledge, infection control performance, and self-efficacy for nursing students. Additionally, the realism of the VRICS program and the students’ level of satisfaction with the program were assessed.t test, paired t test, Mann-Whitney U test, and Wilcoxon matched-pair signed-rank test. The VRICS program consisted of a prebriefing, including direct practice of donning and doffing PPE, VR simulation, and debriefing. The VR simulation comprised 3 sessions: donning and inspection of PPE in the dressing room before entering the negative-pressure isolation unit; assessing for suction care, nasopharyngeal suctioning, and checking of COVID-19 patients in the negative-pressure isolation unit; and doffing PPE in the dressing room. The total execution time for the program was 180 min.This was a quasi-experimental study based on a controlled pretest-posttest design. Third- and fourth-year nursing students were divided into an experimental group (n=25) who participated in a VRICS program and a control group (n=25) with no participation. Data were collected from November 13 to December 10, 2021, and analyzed using descriptive statistics and the z=–3.28, P<.001), infection control performance , and self-efficacy . The experimental group’s mean scores for realistic immersion and learner satisfaction were 4.49 (SD 0.50) points and 4.75 (SD 0.38) points , respectively.Compared with the control group, the experimental group showed significantly greater improvements in PPE knowledge (The VR simulation training program involving pediatric COVID-19 patients combined skills training effectively and enhanced theoretical knowledge, respiratory care skills, and infectious disease preparedness. Thus, it could be applied to training nurses to respond more effectively to public health situations involving infectious diseases, including the COVID-19 pandemic. Health care professionals, including nurses, who care for COVID-19 patients, are at direct or indirect risk of exposure to the virus or other infectious substances. Failure to comply with the appropriate control guidelines can make them a source of infection or a mediator that can spread COVID-19 . Thus, iVR refers to a technology that creates a simulation of real surroundings with people actually experiencing an environment that may be difficult to experience in daily life . VR simuYu et al  developeMeanwhile, prebriefing, with its importance in simulation emphasized recently , is a stHowever, there are limited VR simulation programs on medical conditions, treatment, and patient care for training nurses or nursing students for responding to respiratory infectious disease epidemics. In particular, infection control in hospitals has become more important due to the COVID-19 pandemic and PPE, such gloves, disposable gowns, N95 masks, protective goggles, and shoe covers, can be a primary physical barrier against infection. Despite this, it is difficult to find virtual reality infection control simulation (VRICS) programs applied to pediatric patients for donning PPE, which is not normally used and may appear unfamiliar and complicated. This study thus aims to develop and test a VR simulation program incorporating pediatric COVID-19 cases.The objective of this study was to develop a VRICS program regarding donning and doffing of PPE and respiratory care for pediatric patients admitted to an isolation unit for respiratory infectious disease and apply the program to nursing students to identify the effects of the program on PPE knowledge, infection control performance, and self-efficacy, as well as the realism of the VRICS program and the level of satisfaction with the program. Accordingly, the study established the following hypotheses:Hypothesis 1: The experimental group (participation in the VRICS program) and control group (no participation) will show a difference in PPE knowledge.Hypothesis 2: The experimental and control groups will show a difference in infection control performance.Hypothesis 3: The experimental and control groups will show a difference in self-efficacy.This was a quasi-experimental study based on a controlled pretest-posttest design for the development of a VRICS program with the objective of comparing PPE knowledge, infection control performance, and self-efficacy between the participating and the nonparticipating group of nursing students, as well as to identify the level of satisfaction with the program.The target population was all nursing students in Korea, and the study selected third- and fourth-year nursing students from a nursing college in “J” City in Gyeongsangnam Province. The inclusion criteria consisted of nursing students currently enrolled in a nursing college who have clinical practice experience, completed courses on pediatric nursing and a core basic nursing skill course titled “Practice for Donning and Doffing Standard PPE and Managing Medical Waste” in the regular curriculum, and voluntarily consented to participate in the study. The exclusion criterion was the lack of consent to participate in the study. The sample size for the study was calculated using G*Power version 3.1.9.7 . ConsideThe general characteristics of the participants included age, sex, previous semester grades, and VR experience.PPE knowledge was measured using the tool originally developed by Choi  for PPE Infection control performance was measured based on items related to PPE and pediatric respiratory care. PPE-related items were measured using the tool originally developed by Kwon  for measSelf-efficacy refers to the personal belief about whether something new that has been learned can be applied . Self-efRealistic immersion refers to the perception of presence in an environment that provides realistic illusion . RealistSatisfaction with the VR simulation program was measured after providing the program to the participants using 3 items developed by Yu et al  and modiThe VRICS program for patients with COVID-19 was constructed in the order of prebriefing, including prepractice, VR simulation, and debriefing . This prPrebriefing in this study consisted of standard prebriefing, along with demonstrations and actual practices. The participants received orientation on the goal of the simulation, introduction, review of scenarios, and the simulation lab environment. Subsequently, knowledge transfer learning regarding COVID-19 infection and suction care for preventing droplet transmission and oxygen therapy in a negative-pressure environment for inpatients with COVID-19 was introduced. The participants were instructed to watch an expert role-modeling video on donning and doffing PPE. Each participant practiced donning and doffing PPE for 15 min. During this process, the instructor corrected any errors made by the participants and encouraged the participants to perform the tasks accurately. Moreover, the participants were allowed to take photographs of each other while wearing PPE to experience success. This was to maximize the effects of VR simulation by carrying out learning using the actual experiences of the students. Users unfamiliar with VR technology may need additional time to become familiar with the controller . TherefoThe VR simulation consisted of 3 sessions: (1) donning and inspection of PPE in the dressing room before entering the negative-pressure isolation room , (2) nasTo operate the VR simulation program, an instructor’s computer, the learner’s HMD, and multimedia learning equipment were set up and a VR learning space with a radius of approximately 3 m from the learner was created so that learner movement and depth could be detected, as well as any danger during such movement. In addition, a user interface (UI) was created to induce movement to subsequent steps or to provide brief guidance on some procedural details. Each scenario was carried out for 15 min per participant. The program was designed to allow students who were not taking part in the VR simulation lesson to gain an indirect learning experience by watching the VR screen and other students taking part in the lesson. Students who completed the program were given 20 min to share and write about their opinions about the VR simulation, its strengths and weaknesses, areas of improvement, and when such a program should be operated if it is added to the curriculum. VR simulation took approximately 180 min to complete.Upon completion of the VR simulation, the participants and instructor shared their experiences about the VR simulation for approximately 20 min, discussing similarity to reality, immersion, usefulness, and satisfaction. Moreover, feedback about areas of improvement was exchanged, and the students were given time to describe their thoughts.Data were collected from November 13 to December 10, 2021. Students participating in only the questionnaire survey (control group) and students participating in the VRICS program  were recruited. Participants in the control and experimental groups had already completed the respiratory care for pediatric patients and the core basic nursing skill practice about donning and doffing standard PPE, not including level D PPE, and managing medical waste before this program. All participants completed an online preintervention survey. The control group completed an online postintervention survey 3 weeks after the preintervention survey. For preventing the diffusion of the experiment, the posttest for the control group was administered first. The experimental group completed a written postintervention survey after the completion of the program, and the completed questionnaire was placed in an envelope, which was retrieved by a research assistant. The survey required approximately 10-15 min to complete.t test. The normal distribution of PPE knowledge, infection control performance, and self-efficacy scores was tested using the Shapiro-Wilk test. Normality test results indicated that the experimental group showed a normal distribution, but the control group did not show a normal distribution for PPE knowledge scores, and thus, a homogeneity test was performed using the Mann-Whitney test. Both experimental and control groups showed a normal distribution for infection control performance and self-efficacy scores, and thus, a homogeneity test was performed using an independent t test.Collected data were analyzed using SPSS Statistics version 25.0 (IBM Corp). The general characteristics of the participants were expressed as real numbers and percentages, while homogeneity of the general characteristics between the experimental and control groups was tested using a chi-square test, the Fischer exact test, and an independent U test and the Wilcoxon matched-pair signed-rank test, while differences in infection control performance and self-efficacy between the experimental and control groups were tested using an independent t test and a paired t test.Differences in PPE knowledge between the experimental and control groups were tested using the Mann-Whitney Realism of the program and satisfaction with the program in the experimental group were analyzed by the mean and SD.After obtaining approval from the institutional review board of the institution affiliated with the researchers (GIRB-A21-Y-0061), an online preintervention survey was used to inform the participants about the objective of the study, their right to refuse participation during the study, and the no negative consequence on their school grade even if they did not participate in the study. Further, informed consent was obtained from the participants. Upon completion of the study, each participant was given a coffee gift certificate worth US $10-20.The general characteristics of the experimental and control groups and the PPE knowledge, infection control performance, and self-efficacy scores were homogeneous before the intervention .The results of testing the effects of the VRICS program are shown in z=–3.85, P<.001). In contrast, the control group did not show a significant change in the mean score . The experimental group showed a significant increase in PPE knowledge compared to the control group as a result of the program . Thus, hypothesis 1 was accepted.Hypothesis 1 (difference in PPE knowledge): The experimental group showed a significant increase in the mean score from 14.52 points preintervention to 16.60 points postintervention . In contrast, the control group did not show a significant change in the mean score . The experimental group showed a significant increase in infection control performance compared to the control group as a result of the program . Thus, hypothesis 2 was accepted.Hypothesis 2 (difference in infection control performance): The experimental group showed a significant increase in the mean score from 3.28 points preintervention to 4.69 points postintervention . Meanwhile, the control group showed a decrease in the mean score . The experimental group showed a significant increase in self-efficacy compared to the control group as a result of the program . Thus, hypothesis 3 was accepted.Hypothesis 3 (difference in self-efficacy): The experimental group showed a significant increase in the mean score from 6.04 points preintervention to 6.64 points postintervention  points (out of 5 possible points), and the mean learner satisfaction score was 4.75 (SD 0.38) points (out of 5 possible points); see In this quasi-experimental study, the cases of infection control for pediatric patients admitted to a negative-pressure isolation unit for COVID-19 infection were incorporated into a VR simulation program, and the effects of the program were assessed. Students who participated in the training program were third- and fourth-year nursing students with theoretical and clinical practice experience in pediatric nursing.The findings of this study demonstrated that combining a VR simulation technique and skill training could enhance knowledge and skills for the preparation of respiratory infection care by future nurses. The experimental group, which participated in the VRICS program, showed significantly higher theoretical knowledge about PPE, infection control performance, and self-efficacy than the control group, which did not participate in the program. These findings suggest the need for VR simulation training before future nurses go to the front line to respond to respiratory infectious diseases. This VR simulation training was different than conventional training in that it provided simulated scenes of the isolation unit. Moreover, COVID-19 cases were incorporated into the training program. The order and accuracy of the donning and doffing of PPE and procedures related to respiratory care for pediatric patients were validated by infection control nurses and experts with at least 10 years of clinical experience in a neonatal intensive care unit (ICU) to simulate the actual environment related to COVID-19. Consequently, the participants were able to learn the procedure of donning PPE in the front zone and inspect the donned PPE before entering the isolation unit.The participants were able to check the patient’s name and assess the respiratory status using lung sounds and patient monitoring values for the safety of the patient inside the isolation unit. Subsequently, the participants were able to learn the procedure for suctioning oral and nasal excretions from the pediatric patient, while preventing aerosol transmission. This step included procedures for accurately checking the suction pressure and excretion patterns. The participants also performed procedures for discarding suction tubes and disposable gloves used for respiratory care into the isolation unit medical waste container. They also performed the procedure of exiting the isolation unit and moving to the doffing room before accurately doffing the shoes, gown, gloves, goggles, and mask and discarding them in the waste container inside the contaminated zone. This program simulated the procedures involved in the assessment of clinical symptoms and nursing intervention for patients with COVID-19, contact isolation, and disinfection. The use of PPE can markedly reduce the infection risk associated with caring for patients with COVID-19 ,34. TherWith no signs of the COVID-19 pandemic subsiding worldwide, training for and acquiring of essential infection control skills are important for not only health care professionals but also for nursing students in coping with infectious diseases under such difficult circumstances. In the immersive VR learning environment using VRICS from this study, the participants can learn theories and skills essential for infection control with the incorporation of respiratory care for pediatric patients with COVID-19. Moreover, the VRICS program can reduce the risk of infection and ensure the safety of the trainees by avoiding direct contact with patients with COVID-19. VR simulation also offers the advantage of allowing individualized learning by overcoming the time and space constraints faced by traditional simulation learning during the COVID-19 pandemic, when gathering large groups is difficult.P<.01). These results support the findings of our study showing a significant increase in PPE knowledge and infection control performance in the experimental group. However, technical ability increased significantly in the control group, and there was no difference in the postdisaster management score, which was contradictory to the findings of this study. Meanwhile, a study by Yu et al [t48=–2.16, P=.02) than the control group that received only clinical practice training. These results support the findings of this study showing a significant difference in self-efficacy between the experimental and control groups.A study by Zhang et al  developeYu et al  reportedMeanwhile, prebriefing was a crucial component of simulation learning ,36, and According to Witmer and Singer , presencThe mean learner satisfaction score in this study was 4.75 (SD 0.38) points (out of 5 possible points), similar to 4.79 (SD 0.35) points reported in a study on high-risk neonatal VR simulation by Yu et al . Among tThe participants showed a PPE knowledge score of 14-15 out of 20 points and an infection control performance score of 3 out of 5 points (average performance). Such results indicated that nursing students lacked PPE knowledge. Moreover, the control group, which did not participate in the VRICS program, showed a slight increase in yjr PPE knowledge score and no significant increase in infection control performance after 3 weeks. In the assessment of PPE knowledge using a questionnaire, the scores may have increased due to the participants remembering questions from the preintervention survey or being motivated to correct a wrong answer. No significant change in infection control performance in the control group indicates the need to provide programs or opportunities to teach skills that enhance COVID-19 infection control and allow the students to practice such skills.Zellmer et al  reportedThis study had a few limitations. First, because the study population consisted of nursing students from a single college, caution should be taken when interpreting the results. Each college has different training programs, and students may have different levels of knowledge, performance, and self-efficacy, even if they are in the same grade. Second, the procedures for VR-based skills need to be revised to be more realistic. The VR program used in this study could not reproduce delicate procedures or nursing skills. This is because detailed procedures could not be introduced to each process, since the video run time was set to 15 min to prevent dizziness from wearing the HMD for too long, and allocation of the total operating time for the VR program. Moreover, because learning the skills for donning and doffing of PPE and respiratory care was a bigger goal, the program was set up to complete the applicable skill procedure just by hand touching through hand tracking. Therefore, when developing future VR programs, it is necessary to improve the imaging technology to reproduce even delicate motion. If not, learning effects could also be maximized by including the process of actually donning and doffing PPE, as in this study. The third limitation was testing of skills by means of a self-questionnaire in this study. Therefore, it is suggested that actual performance be evaluated using evaluation mode in a VR program.In this study, a VR simulation training program with the inclusion of infection control cases involving pediatric patients with COVID-19 admitted to an isolation unit was developed, and the effects of the program were tested. For infection control education, combining VR simulation and skills training was more effective than conventional instruction in producing better outcomes. This training program demonstrated more benefits for enhancing theoretical knowledge, respiratory care skills, and infectious disease preparedness, and thus, it could be applied to training nurses to respond even better to public health situations involving infectious diseases, including the COVID-19 pandemic.