the study of epidemics has a long and fascinating history that dates back to daniel bernoulli who modeled the spread of smallpox .theoretical models have been quite successful in describing the spread of infectious diseases .closely related models have been applied to a truly remarkable set of contagious processes including hiv , computer viruses , spread of technological innovations , outbreaks of social and political unrest , and rumor propagation .the convenient deterministic framework is most commonly used to model the spread of an epidemic . in this formulation ,coupled ordinary differential equations describe the evolution of macroscopic properties such as the average number of infected , the average number of recovered , and so on .however , the deterministic approach can not capture fluctuations which are inevitable due to the finite size of the population .in many scenarios such as the spread of an infection in an isolated boarding school with say , a hundred children , statistical fluctuations are certainly significant .the stochastic framework which involves evolution equations for the entire probability distribution is instead required to describe finite populations .theoretical analysis of the stochastic framework is challenging , even for the most basic infection processes , and many questions concerning finite - size effects and extremal properties of the probability distribution functions remain unanswered .  in this paper, we investigate the stochastic version of the classic susceptible - infected - recovered ( sir ) infection process . in this model ,the population consists of susceptible , infected , and recovered individuals , and the infection spreads through contact between infected and susceptible members of the community .each infected individual spreads the infection at a certain rate , denoted by , while infected individuals recover at a rate set to one .the epidemic threshold is . below the threshold ,the infection can not maintain itself ; above the threshold , the infection can take off .we focus on epidemic outbreaks at or near the threshold .such `` threshold epidemics '' are especially interesting . from a theoretical viewpoint ,the infection and the recovery processes are in some sense in perfect balance precisely at the threshold .while most infections are small , large outbreaks may occasionally happen , and hence , threshold epidemics exhibit large fluctuations . from a practical viewpoint , human efforts at disease prevention reduce the infection rate thereby driving infectious diseases from the pandemic to the endemic phase .meanwhile , natural evolution may increase the infection rate of diseases hovering just below the threshold , thereby enhancing the likelihood of an outbreak .these epidemics are subtle : they may be difficult to detect as well as difficult to control .we first consider infinite populations and focus on the behavior precisely at the epidemic threshold .we analyze the rate equation for the probability that the number of infected equals and the number of recovered equals at time .the typical number of infected grows linearly with time while the typical number of recovered grows quadratically with time .we show that the joint distribution function obeys the scaling form we obtain the laplace transform of the scaling function analytically , and present a comprehensive asymptotic analysis .we also discuss scaling properties of the respective single - population distribution functions for the number of infected or the number of removed .next , we consider finite populations .we combine the infinite population results with heuristic arguments to derive scaling laws for the size and duration of outbreaks in a finite population of size . at or near the epidemic threshold, the effective infection rate is reduced because the population is finite .we find that the maximal size of the outbreak , , and the maximal duration of the outbreak , , obey the nontrivial scaling laws we also formulate the range of infection rates near the epidemic threshold for which these scaling laws apply .the rest of this paper is organized as follows .we begin with the definition of the infection process , given in section ii . in section iii ,we treat the infinite population limit by using the rate equation approach .first , we derive the outbreak size distribution .we then compute two single - variable distributions the probability distribution for the number of infected individuals and the probability distribution of the number of recovered individuals .next , we discuss scaling and extremal properties of the joint distribution function . in section iv , we consider finite populations and obtain finite - size scaling laws for the size and duration of the outbreaks .we discuss the results in section v , and the appendix details some necessary inverse laplace transforms .the spread of infectious diseases has been widely studied on regular and irregular spatial structures such as lattices and complex networks .nevertheless , many characteristics of stochastic epidemics , e.g. finite - size properties , remain open questions even for perfectly mixed populations . throughout this paperwe assume that the populations are well - mixed so that any infected individual can spread the disease to any susceptible individuals .  in the ubiquitous susceptible - infected - recovered ( sir ) infection process , the population includes three types of of individuals :  * uninfected individuals who are susceptible to the infection .* infected individuals who are actively spreading the disease . * individuals who are neither infected nor susceptible including those who have been infected and subsequently recovered , or became immune , or removed .an individual may proceed from type to to . in this simplified model, all infected individuals may spread the disease .we investigate the continuous - time version of sir infection process . at any given moment , the population consists of susceptible , infected , and recovered individuals ; the size of the total population , , remains fixed . the sub - populations change due to two competing processes infection and recovery . an infected individual may infect a susceptible one with rate , where is the infection rate : the overall infection rate is proportional to the size of the susceptible population times the size of the infected population .infected individuals recover at a constant rate , without loss of generality , the recovery rate is set to one .we consider the natural initial condition where a single infected individual is embedded in a `` sea '' of susceptible individuals , that is , at time .the infection process ends when no infected individuals remain , .the final number infected , , measures the size of the epidemic outbreak .since the infection process is random , the size of the epidemic outbreak is a stochastic quantity .first , we investigate the distribution of this quantity as its basic characteristics show how the infection process can be in one of two phases : an endemic phase where a microscopic number of individuals are infected and a pandemic phase where a macroscopic number of individuals are infected .  in this sectionwe analyze the infinite population limit , .let be the probability that the size of the epidemic outbreak equals .for the initial condition , the total infection rate and the total recovery rate are and as follows from . in the very first step , the infected individual either recovers , or a second individual become infected .the former happens with probability , while the latter occurs with the complementary probability .hence , the distribution obeys the recursion equation valid for all .the convolution term corresponds to the case where an additional individual become infected in this situation there are two infection processes , and in the infinite population limit these two processes are completely independent . starting with ,the recursion equation gives , , etc .we now introduce the generating function using the generating function we convert the infinite set of recursion equations ( [ gn - eq ] ) into the quadratic equation , . out of the two solutions ,the proper one satisfies the requirement ,   the behavior of in the limit reveals the basic characteristics of the infection process .the probability that only a _finite _ number of individuals are infected is given by the derivation of follows from and .the threshold value separates two regimes of behavior an endemic regime and a pandemic regime . below the threshold ,the number of infected individuals is always finite . above the threshold, there is a finite probability that a finite _ fraction _ of individuals becomes infected , and as a consequence , the average size of the epidemic outbreak is macroscopic , that is , proportional to the size of the total population .also , the average number of infected individuals readily follows from the generating function , viz . .equation gives as expected , the size of the outbreak diverges in the vicinity of the threshold .  to find , we expand the generating function in powers of .the distribution of outbreak size is a product of an exponential and an algebraic factor , expressed as a ratio of gamma functions , ^{n-1}.\ ] ] by using the asymptotic behavior as , we conclude that at the epidemic threshold , , the size distribution has a power - law tail as . for threshold epidemics ,there is balance between infection and recovery , as indicated by . while the majority of outbreaks are small , the algebraic behavior suggests that there is a considerable likelihood for large outbreaks to occur .  in the remainder of this section , we focus on the behavior at the epidemic threshold , .we begin with the probability that there are infected individuals at time .irrespective of the infection rate , this distribution function satisfies a closed evolution equation , and for the critical case , satisfies the initial condition is .equations are closed because in the infinite population limit , , the population of infected individuals does not depend on the populations of susceptible and recovered individuals .we comment that the master equation is a discrete diffusion equation with a diffusion coefficient equal to the size of the infected population .to solve we use the exponential ansatz : for , with the initial conditions and to assure the validity of .this ansatz reduces the infinite set of equations to two ordinary differential equations , and . solving these coupled equations we obtain for , the size distribution is purely exponential .  the quantity gives the probability that the infection process has subsided by time . from and the initial condition , we obtain .equivalently , one can deduce this result from the normalization requirement .note also that the survival probability , defined as the probability that the infection remains active at time , is simply , and hence , it is given by in the long time limit , the survival probability decays algebraically , .interestingly , the first moment of the distribution is conserved , , and in this sense , the competing processes of infection and recovery balance when .hence , if we restrict our attention to active outbreaks , the average number of infected individuals grows linearly with time , , where .  in the long - time limit, the distribution has the scaling form this scaling behavior immediately follows from in the limit and with the scaling variable kept fixed .the master equation is closed since the infected population is decoupled from the other two populations . in contrast , the recovered population is coupled to the infected population .therefore , we must analyze the joint distribution , that is , the probability there are infected and recovered at time .of course , the joint distribution function gives a complete description of the state of the system and for example , .the joint distribution obeys and . in this rate equation , the first gain term accounts for infection , while the second gain term represents recovery . for small and one can compute the joint distribution recursively : , , /2 ] in terms of . in writing, we also used the shorthand notations for example , we can verify that the generating function yields the size distribution of outbreaks . in the long - time limit , the last term on the right - hand - side of vanishes since the quantities and are both divergent . with the shorthand notation , we have . by substituting into, we confirm that when .first , we analyze the probability distribution . by definition, is the probability to have recovered individuals at time . of course ,the distribution of recovered differs from the distribution of infected in that in the long - time limit , it approaches a nonzero value , , whereas for all . to study the long - time asymptotic behavior , we write .we have as .the corresponding generating function is given by we obtain this expression from the joint generating function by setting , where .  according to the scaling behavior , the typical size of the infected population grows linearly with time , .heuristically , since the recovery rate is constant .consequently , the typical size of the recovered population is quadratic , , and hence , we expect the scaling behavior   the laplace transform of the scaling function follows immediately from behavior of the generating function in the limit .we take the limits and while keeping the variable fixed . in this limit, we have and the sum over turns into the integral by using this scaling transformation along with eq ., we find the laplace transform of the scaling function   the inversion of this laplace transform through integration in the complex plane is detailed in the appendix where we show that the scaling function is given by we now substitute this expression into the scaling form and observe that the algebraic factor and the final distribution cancel each other .thus , the distribution has the same scaling form as the distribution , the scaling function is given by the sum the leading asymptotic behaviors of the scaling function are the algebraic behavior in the small- limit is consistent with the power - law tail . to obtain this behavior, we simply convert the sum in into an integral .we note that with the scaling form and the small- divergence , the quantity is indeed finite . also , the first term in the series yields the exponential behavior in the large- limit .hence , both the distribution of recovered and the distribution of infected have exponential tails .we now analyze the joint distribution .we restrict our attention to active epidemics that correspond to .thus , we focus on the following component of the joint generating function this component follows directly from the generating function , , and by using the explicit solution , we obtain }.\end{aligned}\ ] ]  the scaling behaviors and for the single - population distributions imply that the joint distribution function adheres to the scaling form stated in the introduction . since the survival probability decays with time according to, the scaling function is normalized .the above analysis suggests use of the joint laplace transform that is merely the continuous counterpart of the generating function . to obtain from given in , we take the limits , , and while keeping the variables fixed . by taking these limits ,we observe that the right - hand side of becomes , and find the joint laplace transform in explicit form we stress that the quantity describes only active infection processes .one can verify that .moreover , in the limit , we have , for which the inverse laplace transform is immediate , .indeed , we recover the scaling function in eq . .since the inverse laplace transform of with respect to the variable is immediate , we have the appendix outlines how to invert this laplace transform to obtain the leading asymptotic behaviors for large- and small- , & \eta \to 0 , \\\frac{\pi^2\,2^{1/4}}{\xi^{3/4}}\,\,\exp\left[-\pi^2\eta+\pi\sqrt{8\xi\eta}-\xi\right ] & \eta \to \infty .\end{cases}\ ] ] these limiting behaviors apply for a fixed value of .the function captures all moments of the scaling function .for example , by expanding equation as a taylor series in the conjugate variables and , , and comparing with the definition , we obtain the lowest - order moments , , and . in particular , so there is positive correlation between the size of the two populations . intuitively , we expect that long - lasting epidemic outbreaks involve large numbers of infected and recovered , while the opposite is true for short - lived outbreaks .  for completeness , we mention that the above analysis can be repeated for inactive outbreaks . starting with given in and following the steps leading to , we find that the distribution that an epidemic outbreak has ended by time and that the size of the outbreak equals , has the scaling form .the scaling function resembles given in the leading asymptotic behavior in the small- limit is identical to that in , and again , there is exponential decay , albeit with twice the coefficient , in the large- limit .as expected , inactive outbreaks dominate the size distribution of the recovered population at large times .the zeroth and first - order moments of the distribution given in equations and show that the size of the epidemic outbreak is microscopic below the epidemic threshold , , but macroscopic above the threshold , .these results hold for infinite populations , yet real - world epidemic outbreaks involve finite populations , and there is practical need for understanding how basic characteristics such as the average size and the average duration of the epidemic depend on the size of the population , .finite - size effects are most pronounced in the vicinity of the epidemic threshold .let us consider large but finite populations , , and let us consider a scenario in which the susceptible population has been depleted by , that is , .from the very definition of the infection process , we determine that the infection rate is reduced , , with because the total population is finite . clearly , as the large `` reservoir '' of susceptible individuals shrinks , the infection process weakens .  ) . shownare monte carlo simulation results representing an average over independent realizations of the infection process ( circles ) .a line of slope is also shown as a reference.,scaledwidth=45.0% ]  let us now consider the average size of the outbreak for a threshold epidemic ( ) .we assume that there is a maximal outbreak size , and that the outbreak can not exceed this size due to depletion in the number of susceptible individuals . on the one hand , equations and suggest that . on the other hand, the algebraic distribution gives a second estimate for the average size of the outbreak .equating these two estimates , , we conclude that as stated in .while a naive interpretation of the effective infection rate would suggest that the cutoff is proportional to the total population , we find that threshold epidemics have a substantially smaller upper bound .realizations ( circles ) are compared with the reference line ., scaledwidth=45.0% ]  using and we see that the average size of a threshold epidemic grows as this interesting scaling law was established by martin - lf and it has been confirmed in several recent studies .figure [ fig - nav ] shows a numerical verification of this behavior . for finite populations , this scaling law holds in the neighborhood ( often termed critical or scaling windows ) of the epidemic threshold . to estimate the size of this neighborhood, we simply compare with .the size of the outbreak grows sub - linearly in the population size as long as latexmath:[\[\label{neighborhood }  size of the population grows . yet , for moderate populations , the size of this neighborhood is considerable .the scaling law indicates that for finite populations , there are actually three regimes of behavior . well below the epidemic threshold , a finite _ number _ of individuals becomes infected .well above the threshold , a finite _ fraction _ of the population becomes infected . in the vicinity of the threshold , the size of the outbreak grows as the power of the population size , with the scaling variable .a finite value of indicates that the infection rate is in the neighborhood of the threshold .( the scaling function can be extracted from refs . . )the duration of a threshold epidemic also adheres to a finite - size scaling law .since the size of the recovered population grows quadratically with time , , as implied for example by the scaling behavior , we conclude that there is a maximal time scale , and that the duration of a near - threshold outbreak may not exceed this scale .we can estimate the time scale stated in , from the heuristic argument .the average duration of the infection process follows from the survival probability , , and using , we find that the average duration grows logarithmically with time ( fig .[ fig - tav ] ) , logarithmic growth , albeit with a unit prefactor , was predicted by ridler - rowe .  the scales and fully characterize the size and duration of the infection process when . for example , the survival probability to have at least one infected at time obeys ( fig .[ fig - scal ] ) .however , the simulations show that the convergence to this scaling form is not uniform it is slow for short durations but fast at large durations .   versus the normalized duration time .the data corresponds to an average over realizations.,scaledwidth=45.0% ]  we performed numerical simulations to check the scaling predictions for the case .our numerical method is merely a monte carlo procedure to solve the master equation .we conveniently keep track of two populations , and , from which the overall infection rate and recovery rate are respectively and .the probabilities and that the following step involves infection or recovery are calculated and then , the populations are updated accordingly , the infection process ends when for the first time .time is updated by the inverse of the total rate . with this straightforward numerical procedure, we can simulate as many as independent runs in populations as large as . as shown in figures [ fig - nav ] and [ fig - tav ] ,there is excellent agreement between the numerical results and the theoretical predictions for the average size and the average duration .we investigated various statistical properties of the sir infection process at a threshold infection rate .we analyzed the rate equations for the two - population distribution function that characterizes the probability that the system has a specified number of infected and recovered individuals .our analysis yields scaling behavior in the asymptotic long - time limit and gives extremal properties of the joint distribution function as well as the associated single - population distributions .we used these infinite population results to justify scaling laws for finite populations .outbreaks in the vicinity of the epidemic threshold have a distinct size , characterized by non - trivial power - law dependence on the total population size .this scaling behavior applies in the vicinity of the epidemic threshold .the size of this neighborhood , , is larger than the canonical value expected from the traditional large - size expansions or from the deterministic description .therefore , statistical fluctuations and finite population effects are pronounced and may be subtle near the epidemic threshold .we note the an identical scaling arises near the percolation point for erds - rnyi random graphs .additional connections between sir infection processes and random graphs were reported recently .the scaling laws for the time dependence and the size dependence are useful . for example , the scaling laws for the critical kinetics can be used to find the epidemic threshold in numerical simulations of infection processes on complex networks for which the threshold is not known analytically .furthermore , the number of coupled ordinary equations needed to compute the joint distribution numerically is in principle quadratic with .however , the scaling laws and imply that the relevant number of equations is much smaller , being proportional to .  for large but finite populations , we understand the basic scaling laws , but much less is known about finite - 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Abstract: we study the classic susceptible - infected - recovered ( sir ) model for the spread of an infectious disease . in this stochastic process , there are two competing mechanism : infection and recovery . susceptible individuals may contract the disease from infected individuals , while infected ones recover from the disease at a constant rate and are never infected again . our focus is the behavior at the epidemic threshold where the rates of the infection and recovery processes balance . in the infinite population limit , we establish analytically scaling rules for the time - dependent distribution functions that characterize the sizes of the infected and the recovered sub - populations . using heuristic arguments , we also obtain scaling laws for the size and duration of the epidemic outbreaks as a function of the total population . we perform numerical simulations to verify the scaling predictions and discuss the consequences of these scaling laws for near - threshold epidemic outbreaks .


the device - to - device ( d2d ) communication paradigm is expected to become the key enabler for propelling local , proximity - based communications in future wireless networks . in this communication type , two closely located user equipments ( ues )are eligible to bypass the cellular base station ( bs ) and effectively communicate in a direct mode .the integration of d2d in emerging 5 g systems is expected to be beneficial for mobile stakeholders , mainly by unlocking new applications related to services based on proximity , vehicular - to - vehicular ( v2v ) communications and other commercial opportunities that d2d can offer .in addition to that , there are also radical benefits in the overall network operation : first , the ues proximity entails increased data rate performance , low energy consumption as well as low transmission latency .also , the reuse ability of the cellular and d2d links to simultaneously utilize the same radio resources translates to noteworthy spectral efficiency .thus , because of the limited cellular resources and the ongoing proliferation of simultaneous data requests that need to be satisfied , the underlay notion of d2d communication is preferred due to its resource efficiency attribute .  in parallel ,the virtualization of wireless radio resources has arisen as a promising solution to encounter the ongoing increasing data demand in today s and emerging future networks .wireless resource virtualization ( wrv ) is currently emerging as a disruptive technology that offers significant benefits to different networks and service providers ( n&sps ) as well as enabling vertical industries to create their own wireless network .briefly , other than the fact that co - existing networks , which will be called tenants in the sequel , are able to share the substrate physical infrastructure that entails reduced capital and operational expenditures ( capex and opex , respectively ) , wrv ameliorates the utilization of radio resources via sharing them among the different n&sps .hence , the exploitation of the wireless virtualization and network programmability merits on top of the integration of d2d paradigm can lead to improved network performance in terms of spectrum efficiency as well as overall network performance .however , the advent of the data - driven era brings in a number of challenges mainly due to the resulting cell and user densification . among all , a prevalent problem that is expected to attract not only academic but also industrial interest is that of direct communication between users that are subscribed to different mobile network operators ( mnos ) .a solution to this problem can create a fertile ground for introducing new business models that will fully leverage the d2d potentials .technically , the weight should be primarily put on defining how the involved mnos will coordinate their spectrum to satisfy their subscribers quality of service ( qos ) requirements . in the case of single mno, underlaying d2d links are allowed to utilize the licensed cellular spectrum that is provided by the operator . on the other hand , in the case where two devices belong to two different operators, it needs to be decided which resources from which mno will be utilized to realize the d2d connection .therefore , the principal aim is to support a significant number of direct connections along different network operators while at the same time respect the performance of cellular users as well as the overall welfare of the system .  in the next subsection , we delve further into the above issues as with respect to resource sharing across different tenants of the substrate infrastructure .   in envisioned future virtualized and programmable 5 g wireless network architectures , different tenants ( virtual network providers ) will be sharing the physical ( substrate ) network resources using a combination of software defined networking ( sdn ) and network function virtualization ( nfv ) architectures .the core idea behind nfv is to capitalize on virtualization technologies to decouple physical network equipment from the services or functions that run on top of them . under the nfv framework, a network service can be decomposed into a set of virtual network functions ( vnfs ) which are implemented in software and are able to run in general purposed hardware where they can be dispatched on demand .an overview of the nfv architecture is shown in figure [ fig : nfv ] which conforms to the etsi nfv framework .as depicted in this figure , a service request will be handled by the orchestrator which will then inform the virtual function manager about which vnfs are required to be activated for this specific service , whereas the actual physical resources for the vnfs will be handled by the virtualization infrastructure manager ( vim ) .the above defined policies for the service creation will be distributed using the sdn controller ( based for example on openflow ) .  since each tenantwill be allocated a slice of the available network resources ( including also spectrum ) , mobile users that will require device - to - device communication from different tenants will be allocated resources ( resource blocks ) from the device which is originating the communication .this operation might lead to inefficient usage of the tenant s available resources in the long run .we therefore propose the use of an inter - slice coordinator that will allow for optimal usage of multiple tenants resources in the case where the communication is taking place between users subscribed to different tenants .an illustrative example of an inter - slice controller is shown in figure [ fig : cross_tenant ] , for the case of two tenants .such cross - tenant orchestration would allow a more efficient use of the available physical resources per tenant .the concept of inter - slice coordination is being developed within the eu 5g - ppp 5g - norma project where the key motivation is to replace single ran s networked entities by a network slice with a graph of programmable network functions sino - europe 5 g workshop _, november 2015 , beijing , china ] .a cross - tenant controller should be a trusted entity since in order to optimize the overall performance tenants will have to provide intra - slice topological information to the controller which might include , inter alia , the number and location of users in each slice .depending on the actual implementation , the cross - tenant controller can be considered as a broker that runs by the substrate network provider , which can be deemed as a trusted element .note , there is an incentive for all tenants to cooperate since overall network performance is increased ; however , trusted entities in virtualized architectures is a topic well beyond the scope area of this paper .based on that fundamental assumption , the aim of this paper is to quantify the potential achievable gains enabled by such a cross - slice controller .     in this work, we assume that separate network slices allow for multi - tenant d2d discovery and session initiation procedures with similar techniques as the ones defined in , where a protocol is designed to permit the inter - operator d2d communication . without loss of generality , we consider two tenants and for each tenant we assume a number of subscribed cellular and d2d ues ( cues and dues , respectively ) , randomly distributed in a typical hexagonal cell layout .each due is considered to be communicating in a maximum allowed distance with a peer that belongs to different tenant .based on this topological modelling , we propose an integer linear programming ( ilp ) optimization framework that aims at maximizing the sum - rate performance of the involved inter - tenant d2d links while retaining the cellular ues qos requirements of the involved tenants above a predefined performance threshold . to the best of our knowledge ,this is the first paper that deals with the inter - tenant d2d communication optimization in virtualization - enabled networks .the structure of this paper is as follows : section [ sec : literature ] quotes a number of significant , closely related work on the issue of d2d virtualization . in section [ sec : sys_model ] , the system model that considers the integration of inter - operator ( inter - tenant ) d2d communicationsis initially described .then , we proceed by formulating the sum - rate maximization problem for this communication paradigm .section [ sec : results ] illustrates the performance that is achieved based on this optimization methodology and compared to legacy techniques .finally , concluding remarks are provided in section [ sec : conclusion ] .in order to efficiently leverage the radical merits that d2d offers , the interference patterns that are being developed due to the resource reuse need to be limited . to this direction , several works within the literature have been elaborating on devising efficient resource allocation and mode selection techniques for d2d communications that mainly aim at improving network throughput as well as spectrum efficiency .optimal usage of the available radio resources is a well - known nonlinear np - hard problem that becomes even more complex with the integration of d2d communications that underlay the cellular network . for this reason, a common practice has been to devise low - complexity , heuristic algorithmic solutions or to relax part of the constraints ( e.g. power or resource block allocation related ) and propose sub - optimal techniques for d2d - aware resource allocation , .so far , the problem of inter - operator d2d communication is inadequately explored , thus it needs to be carefully encountered in order to harvest the business dynamics of this specific communication type . to the best of our knowledge ,the only existing work on this particular topic is .therein , the authors propose the allocation of inter - operator d2d communications over dedicated licensed radio resources ( overlay d2d ) which the different operators have to negotiate between each other about the amount of spectrum that they will finally dedicate .they formulate this problem as a game between two distinct mobile network operators and decide about the offered spectrum with a best response method that runs in a sequential manner .compared to it , our approach differs in that we consider a virtualized ran infrastructure where inter - operator d2d links can utilize the whole available spectrum ( underlay as opposed to overlay ) to achieve efficient resource sharing other than sum - rate maximization .however , even the dynamics of intra - operator d2d communications in ran virtualized ecosystems are barely explored in up - to - date literature .one of the first efforts towards this direction is the work in , where the authors address the problem of network state information ( nsi ) imperfectness in virtual wireless networks and resource allocation for the software - defined d2d connections .they devise a discrete stochastic optimization formulation to the problem of resource sharing given imperfect nsi and , then , proceed with the introduction of stochastic approximation algorithms for both static and varying channels resource manageability .further , our work in considered the virtualization of the resources offered by different mobile virtual network operators ( mvnos ) in order to support and improve the performance of intra - mvno d2d connections in the uplink scenario .the problem was formulated as an ilp sum - rate maximization problem , based on the constraint that the allocated resources per d2d should be contiguous .heuristic proposed methods were also included as low - complexity solutions .virtualization of the core as well as the radio access network is envisioned as the de - facto way forward for 5 g networks since it can provide higher degree of flexibility to the mobile network operator , whilst with a careful design it can reduce overall network cost , .a preliminary study as with respect to use cases and requirements has also been defined within the 3gpp .also , important architectural aspects have also been discussed in order to support such advanced mechanisms .finally , under the assumption of a virtualized mobile network , the work in considers the issue of resource allocation for d2d nodes via a non - linear optimization framework but does not consider the issue of inter - tenant d2d resource optimization .the studied hexagonal cell area consists of a center - located ran virtualized bs equipped with omni - directional antennas and a number of cellular users and d2d links uniformly distributed .part of the distributed cellular and d2d users are assumed to belong to a specific tenant and are served by its designated slice , whereas the rest of them are subscribed to a second tenant , hence , a separate slice is dedicated to serve them .note that , without loss of generality , we hereafter assume the existence of two tenants .figure [ fig : topology ] depicts the described scenario where intra and inter - slice / tenant d2d communications can take place .the establishment of d2d communication is out of the scope of this paper .briefly , intra - operator d2d session setup is carried out by the session initiation protocol ( sip ) discussed in , whereas the establishment and realization of the inter - operator d2d connection is detailed in .as already mentioned in subsection [ subsec : nfv ] , each tenant is assigned with a slice that will provide , inter alia , spectrum allocation in order to fulfill the expected demand from the serving users .quantitative , this translates to a number of resource blocks ( rbs ) which constitute the available resource pool of the users that are subscribed to specific tenant . considering the legacy procedure , in order to support a d2d link between two users ( intra or inter - operator ) , the resources used for it are allocated only from the rb pool that corresponds to the user that inaugurates the direct communication .however , in this work , we leverage the ability of an inter - slice manager to fuse the available rb pools of multiple tenants in order to enlarge the spectrum availability for inter - operator d2d links ( figure [ fig : cross_tenant ] ) .  following the principles of d2d cellular spectrum reusability in the underlay notion , d2d users are able to utilize the resources of multiple cellular users simultaneously .however , it is important to ensure that the cellular transmissions whose resources are being reused by a d2d pair satisfy their qos minimum requirements .for ease of comprehension , we presume that cellular users primarily occupy one but orthogonal radio resource based on the lte specifications .the reason why we consider this multi - tenant unified rb pool is to increase the resource efficiency for cross - tenant d2d links which are expected to be a significant part of future network connections .this will not only lead to effective usage of the available spectrum , but also improve the overall network performance by potentially increasing throughput and reducing interference . to this direction ,an integer linear programming ( ilp ) optimization solution is proposed to maximize the sum - rate for inter - slice d2d links by respecting at the same time the cellular transmissions performance not to degrade below a predefined threshold .before we formulate the d2d sum - rate optimization problem , the following sets need to be defined :  * is the set of distributed cross - tenant d2d links ; .* is the set of cellular ues ; . * is the set of tenants ; . * is the set of available resources ; . contains all the cellular users that belong to different tenants and are consequently served by separate slices it can be represented as follows : , where .similarly , regarding the fused set of resource blocks ( rbs ) , it consists of all tenants radio resources , so it can be written as , where .  further , we need to introduce the binary decision variable that indicates if a d2d link utilizes a specific rb that belongs to one of the tenants available resource pool . this can be mathematically represented as :   we further proceed with some important system model admissions to pave the way for the problem formulation .first , the path - loss is modeled as follows : for d2d pairs and cellular users , respectively , .parameter stands for the euclidean distance and is expressed in kilometers ( km ) . additionally , the signal - to - interference - plus - noise - ratio ( sinr ) at the d2d receiver of link that uses rb of tenant needs to be satisfied .if we denote by this value which translates for the receiver s need to correctly decode transmitted packets , this constraint can be practically expressed as follows : where is the link gain ( path - loss and slow fading dependent ) of the d2d pair , and is the transmission power of the d2d transmitter over this rb . in the denominator , expresses the link gain between the transmitting cellular user equipment ( cue ) and the receiver of d2d link when using rb of tenant ( cue - d2d interference will be developed when and ) . as it will be mentioned in the sequel , we consider that d2d links will be using orthogonal rbs among each other , i.e. the received interference of the d2d links will be only deriving from cellular ues ( and vice versa ) .lastly , denotes the lump sum power of background / thermal noise and the co - channel interference from other cells ( if existent ) . in that case , we assume that inter - cell interference can be controlled via the application of powerful inter - cell interference cancellation ( icic ) techniques , thus , we are focusing on a single - cell scenario ( ) where the main part of interference ( i.e. intra - cell ) is effectively captured .the sinr threshold ( ) needs to be also satisfied for the cellular transmissions that utilize the same rb ( e.g. ) with a d2d link during the uplink session .this constraint can be written as follows : where is the transmission power of a cellular user , is the link gain between the cue that belongs to tenant and its associated bs when using rb , whereas accounts for the link gain between the d2d transmitter of link and the bs that transmit / receive over the same channel .  consideringthe above definitions , the achievable rate for d2d link that utilizes rb of tenant can be calculated according to the well - known shannon capacity formula : where is the lte - based resource block bandwidth ( 180 khz ) and is expressed in power ratio .lastly , even though we focus on the uplink scenario where communications happen according to the sc - fdma principles , we herein consider that the rbs allocated per user can be non - adjacent ( ) as the evolution of lte towards 5 g systems will eventually enable fully non - contiguous allocation . considering this, we will practically provide an upper bound of the d2d - based rate performance . following the previous admissions , the sum - rate maximization problem for cross - tenant d2d communicationscan be formulated as follows :  [ eq : optimization ]   constraint ensures that each cellular transmission s sinr does nt fall below a predefined value , whereas guarantees the minimum rate requirement for each d2d link . constraints andaccount for the radio resource allocation of each d2d link ; the former ensures that each d2d pair will be assigned with at least one rb to satisfy its transmission needs , whereas the latter upper bounds the resources used by each link to to avoid any resource deficiency for some d2d ues ( dues ) .then , the restriction that each rb can be used by only one d2d link is realized by .finally , denotes the binary nature of the decision variable .finally , it is obvious that the overall rate achieved by a d2d link is and depends on the value assignment of the decision vector that solves this optimization problem .in this section , a set of evaluation results is provided to shed light on the performance of the proposed inter - tenant d2d sum - rate optimization problem compared to legacy approaches and heuristic solutions .   in this subsection ,a number of different d2d - based resource allocation techniques for inter - tenant communications are briefly described .these techniques constitute the compared methodology through which the results to follow are produced .  1 .* inter - tenant optimal * : the proposed method was detailed in subsection [ subsec : problem_def ] .as previously explained , it yields optimal sum - rate performance for inter - tenant d2d users via a powerful ilp solution that virtually fuses the provided to the tenants rb pools and orchestrates the links resource assignment .* inter - tenant heuristic * : complementary to the previous technique , a heuristic algorithm is proposed to seek for a low - complexity , near optimal solution for d2d users that belong to different tenants .one of its chief characteristics is that it tries to achieve a fairly balanced , inter - slice resource allocation by sequentially running for d2d receivers that belong to different tenants .its resource assignment rationale is based on allocating the resource blocks that provide the best channel conditions to each d2d link ( in a sorted way ) following the aforementioned sequential mode . herein , it has to be noted that in order for some rb to be assigned to a d2d link , first , it must be an unallocated one ( among d2ds ) and second , not to degrade the performance of the cellular uplink transmission that utilizes the same rb .then , the algorithm iterates over and over , until one of the following conditions is violated : ( i ) all d2d users reach their upper rb usage limit ( i.e. used rbs ) , ( ii ) the fused rb pool is fully utilized by the active d2d transmissions , or ( iii ) dues sinr requirements over the remaining rbs are not satisfied .the explained method is outlined in algorithm [ alg : heuristic ] .* intra - tenant optimal * : with this technique , problem is decoupled into two separate resource allocation problems for the two different tenants .this means that each tenant solves separately the sum - rate optimization problem for its subscribed d2d users that initiate direct peer communications , based on its corresponding dedicated slice resources . due to the restricted rb availability for the different tenants , this method , even though it is able to provide optimal sum - rate performance from each tenant s side , it is expected to provide a sub - optimal solution in overall .* intra - tenant heuristic * : depending on the number of dues that are subscribed to a specific tenant and initiate a number of inter - slice connections , this tenant is the one to provide the corresponding direct communications with the suitable rb pool to satisfy their transmission needs . to this end , each one of the tenants allocates resources to the corresponding d2ds in a greedy and sorted manner according to best - given channel conditions .this method is similar to the inter - tenant heuristic approach but again is decoupled as it needs to be solved by each different tenant for the subscribed users . * assumptions * : +  * each is assumed to be served by a specific tenant .* orthogonal , round - robin based resource allocation for each is applied . * inter - tenant d2d resource allocation steps * : + step 1 : _d2d rb assignment _+  + step 2 : _ sum - rate estimation _ +  ;   is the achieved rate .[ alg : heuristic ]   the considered system was modeled in matlab , following the lte - a milestones and corresponding network parameters and standards .all the produced results derived after averaging over 1000 monte carlo simulations which have been executed on a _intel(r ) core(tm ) i7 - 6500 at 2.50 ghz and 8 gb ram _ machine .  regarding the topology ,it consists of a hexagonal single cell with randomly distributed cellular users and d2d links .each d2d link consists of two users that are assumed to belong to different tenants . also , the number of cellular users being subscribed to different tenants is varying and considers the tenants disparities in terms of number of subscriptions . without loss of generality , two tenants ,_ a _ and _ b _ , with separate slices are considered .all simulation parameters are listed in table [ tab : sim_parameters ] .lll * parameter * & * value * + cell layout & hexagonal grid + number of tenants ( ) & + cues - d2ds distribution & uniform + macro cell radius & m + maximum d2d link range & m + number of cues ( ) & 50 + number of d2d links ( ) & [ 10,40 ] + d2d path - loss model & + cue - bs path - loss model & + maximum cues power & dbm + maximum dues power & dbm + maximum number of rbs ( ) & + shadowing standard deviation & db + noise power spectral density & dbm / hz + system bandwidth ( ) & mhz +   due to the load discrepancies and divergent number of subscriptions that might characterize the two or more tenants ( either mvnos or mnos ) , the slice that each one is assigned with is expected to be different ( i.e. , having heterogeneous slices ) .popular tenants can be normally allocated with more resources to serve the high number of subscribed users compared to less popular ones . to this end, we consider the case where a popular tenant ( hereafter denoted as tenant _ a _ ) is allocated with double - sized rb pool to serve its subscribers .initially , we assume that for both tenants all the radio resources are occupied by a number of cellular ues according to a round robin scheduling .figure [ fig:1 ] depicts the sum - rate performance for inter - tenant ( inter - slice ) d2d links in relation to varying number of them .on average , almost 11.3% sum - rate gain is achieved by making use of the fused rb pools of the two tenants ( inter - tenant optimal ) compared to the case that a d2d link can be assigned resources only from the resource pool that belongs to the slice to which the user that initiates the direct communication is subscribed ( intra - tenant optimal ) .the maximum performance gap among the illustrated scenarios is met in the case of 16 d2d links , where 12.5% higher sum - rate is achieved with the inter - tenant optimization solution .further , compared to the heuristic inter - tenant approach , the optimal solution is averagely 8.45% better and gradually behaves better with the increase of inter - tenant d2d links number . also , the intra - tenant heuristic algorithm falls short compared to the above - mentioned approaches and it exhibits a maximum of more than 18% sum - rate degradation in comparison to the optimal solution .last , for all the considered approaches , the sum - rate drop that is observed in the two last cases ( i.e. , 30 and 40 d2d links ) is explained by the interference increase to / from cues , as the resource availability gets more restricted .considering the same case study , the cumulative distribution function ( cdf ) of the achieved sinr values for the inter - tenant d2d links is represented in figure [ fig:1c ] .indicatively , in the percentile , the inter - tenant optimal solution s sinr for d2ds is 29.1 dbs , whereas the corresponding values for the inter - tenant heuristic , the intra - tenant optimization and the intra - tenant heuristic are 26 , 25.7 and 23.8 dbs , respectively .this can be interpreted as more than 2 times higher sinr power ratio compared to the inter - tenant heuristic approach .last , the intra - tenant optimal and intra - tenant heuristic methods estimated sinr is almost 2.2 and 3.4 times worse compared to the optimal value .then , we consider the scenario where the two tenants are characterized by the same rb availability but with different utilization levels ( active cellular transmissions per case ) . to this direction , tenant s _a _ radio resources are supposed to be fully occupied by its subscribed cues , whereas tenant s _ b _ resource block availability ranges from 20% to 100% .figure [ fig:2 ] depicts the sum - rate performance of all compared methods in relation to the normalized resource utilization of tenant _ b_. as expected , while the resource occupancy increases , the sum - rate decreases for all methods as new interference patterns between cellular and d2d users arise .however , the performance gap between the inter - tenant optimal and heuristic technique lessens with the increase of the rb occupation levels for tenant _ b _ , as opposed to the rest of the methods where the gap slightly increases . quantifying the above observations, the inter - tenant proposed optimization formulation outperforms the rest of the algorithms in an average of almost 6.5% , 12% and 17.5% , respectively .when both tenants rb pools are fully utilized ( reaching 100% of resource utilization ) , all d2d links are reusing part of the cellular spectrum that cues occupy . in that case, the inter - tenant optimal solution achieves its peak sum - rate gain compared to the intra - tenant methodology ; a 14.3% improvement is observed over the intra - tenant optimal and 18.7% over the related heuristic , respectively .this result can be deemed as highly interesting because the maximum gains take place when needed , i.e. , during network congestion episodes .[ tab : runtimes ]  [ tab : run_times ]  [ cols="<,<,<,<,<",options="header " , ]   further , the sum - rate performance of d2d users in relation to maximum link length is evaluated . by increasing the maximum allowable d2d link length , the sum - rate performance of all compared methods follows a decreasing trend .this is expected , as the increase of d2d link length implies higher sinr degradation not only due to path - loss and shadowing effects but also due to different emerging interference patterns .the performance gains become more clear for the largest values of link lengths . therein , the inter - tenant optimization problem is almost 8.15% , 18.9% and 27.4% better in terms of sum - rate when compared to inter - tenant heuristic , intra - tenant optimal and heuristic solutions , respectively . also , on average , the inter - tenant heuristic method is the one that provides again the closest among all performance as it falls short almost 6% in terms of throughput compared to the optimal one . considering the rest, the optimal solution provides an average gain of almost 16% and more than 22% compared to the intra - tenant optimal and heuristic techniques .finally , in order to give a glimpse of their computational complexity , the running times of the aforementioned algorithms are listed in table 2 .it is shown that with the increase of the number of d2d inter - tenant links , the running times of all compared algorithms decrease .this can be explained by the fact that when less d2d links exist in the topology , the probability that all or many of them will utilize the maximum assignable number of resources ( ) - in order to increase as much as possible their rate performance - raises .consequently , the number of combinations for the orthogonal assignment of d2d links increases .although the inter - tenant optimal solution is proven to be the most complex , its running time remains in acceptable levels .d2d communications are expected to play a key role in emerging 5 g wireless networks by unlocking a plethora of new proximity based applications while preserving the scarce wireless resources for the cellular users .despite the fact that d2d communications received significant attention there has been limited attention to the problem where the two ues of a d2d communication belong to different service and/or network providers . under the assumption of full virtualized core / access networks, the paper provides an optimization framework for efficiently use of virtualized resources across different tenants enabled by a cross - tenant controller . via a wide set of numerical investigationsit has been shown that significant throughput gains of over 10% compared to legacy solutions can be achieved for inter - tenant d2d communications .these results also reinforce the need for implementing a cross - slice coordinator , which can be considered as an extension to sdn / nfv frameworks , in order to efficiently utilize the scarce wireless resources across separate network slices . 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Abstract: device - to - device ( d2d ) communication is expected to enable a number of new services and applications in future mobile networks and has attracted significant research interest over the last few years . remarkably , little attention has been placed on the issue of d2d communication for users belonging to different operators . in this paper , we focus on this aspect for d2d users that belong to different tenants ( virtual network operators ) , assuming virtualized and programmable future 5 g wireless networks . under the assumption of a cross - tenant orchestrator , we show that significant gains can be achieved in terms of network performance by optimizing resource sharing from the different tenants , i.e. , slices of the substrate physical network topology . to this end , a sum - rate optimization framework is proposed for optimal sharing of the virtualized resources . via a wide site of numerical investigations , we prove the efficacy of the proposed solution and the achievable gains compared to legacy approaches .