the join - idle - queue ( jiq ) algorithm is an approach designed to effectively approximate join - the - shortest - queue ( jsq ) algorithm in large - scale distributed systems . in this setting jobs coming into a systemare sent to one of a group of _ dispatchers _ ; the dispatchers are in turn responsible for sending the job to a _ server _ within a cluster of parallel servers for processing . for the join - the - shortest - queue algorithm, each of the dispatchers would have to keep up - to - date load information on all of the servers , which could be undesirable because of the required communication .for the join - idle - queue algorithm , a server informs a dispatcher when it is idle , having completed all of its jobs .the dispatchers each keep a queue of idle servers , dubbed _i - queues _ to distinguish them from queues for jobs .when a job arrives , it sends the job to an idle server from its i - queue , or to a server chosen uniformly at random if its i - queue is empty .further details on the model are given in section [ sec : model ] .the goal of jiq is to provide excellent load balancing most jobs are immediately sent to an idle queue while providing scalability for systems of thousands of servers or larger . in particular , the communication requirements of jiq are low , and importantly the task of determining the load of various servers is not on the critical path when assigning a job . as noted in , alternative schemes requiring a job to communicate to determine the load of servers on the arrival of the job can increase the overall response time , and hence may not be suitable for many practical systems . on the theoretical side , the two - layer approach of load balancing idle servers among dispatchers to allow jobs to be load balanced among servers provides interesting challenges for analysis .jiq systems were introduced in , and analyzed in the large system limit where the number of servers grows to infinity and the ratio between the number of servers and the number of dispatchers is kept fixed .the analysis presented there , however , assumes that there are only idle processors in the dispatchers i - queues .this assumption is inaccurate ; indeed , as noted in , it `` is violated when an idle server receives a random arrival . ''they argue that idle servers receive such arrivals relatively rarely , and show empirically that their approximation appears accurate ( under sufficiently low arrival rates ) .  herewe provide a more direct mean - field fluid limit approach that provides an accurate analysis of the large system limit without the assumptions of the previous analysis by deriving a family of differential equations that corresponds to the behavior large system limit .besides yielding a proper analysis of the basic jiq system analyzed in , this approach allows straightforward analysis of many generalizations .for example , under high loads , it may be better to have a server be placed on an i - queue when it has one job left ( or , more generally , jobs left ) , rather than waiting for the server to become idle , since it may take some time for the i - queue to present the server with a new job . in is left as an open question to analyze such variations .we generalize our equations to this setting here , as well as other variations , such as using last - come first - served instead of first - come first - served at the i - queues in order to determine which server to send a job .( in contrast to the analysis of , we find once simplifying assumptions are removed , such choices can make a small but noticeable difference . )  finally , in both our theory and our empirical work , we examine some potential pitfalls of jiq systems . in particular , under high load , there can be insufficiently many idle queues available , causing degradations in performance as the dispatchers distribute jobs randomly . even when the expected waiting time remains smaller than alternatives , jiq systems can exhibit significantly higher variance , which may be unsuitable for some workloads . as we show later in the paper , having a server place itself on an i - queue when it has one job left greatly mitigates this problem , and we would recommend this variation when there is any possibility that the system may enter a highly loaded state .the main limitation of our analysis is that it applies to the setting of exponentially distributed service times and a poisson arrival process .while generalizations to other service times and arrival processes are theoretically possible by representing them as mixtures of exponentials ( see e.g. ) , or by considering more general partial differential equations ( see e.g. ) a more general analysis of jiq for more general service and arrival processes remains an interesting problem .  also , we emphasize that here our goal is to derive the appropriate equations , examine implications for the use of jiq systems and their variations , and demonstrate empirically the accuracy of this approach .one can also formally prove convergence mathematically , but this is not our main goal .there appears to be no impediment to utilizing standard techniques for mean field fluid limit analysis of queueing systems ( as in ) for jiq systems , so we leave this aspect to the inclined reader .here we follow much of the model and notation of .we have a system with servers and dispatchers ( or equivalently i - queues ) , where we fix the ratio and let .jobs arrive as a poisson process of total rate , where , and are sent to a dispatcher chosen uniformly at random , so at each dispatcher jobs arrive as a poisson process of rate . the service times are exponentially distributed with mean 1 .each dispatcher has an i - queue , which is an ordered list of servers .if the i - queue is non - empty , the dispatcher sends the arriving job to a server on the i - queue ; we start by considering the i - queues being governed by the first - come first - served ( fcfs ) discipline . if the i - queue is empty , the dispatcher sends the job to a server chosen uniformly at random .a server is placed on an i - queue when it finishes all of its jobs ( so its job queue is empty ) and it is not already on an i - queue , so that at any time it is at most on only a single i - queue .we are assuming here that the dispatcher notes when sending a job whether it was randomly assigned from an empty i - queue , so that a server can track when it is removed from an i - queue , but such an implementation is straightforward .there are multiple ways in which a server can choose which i - queue to join . in ,two natural variations are studied .first , on each completion a server can join an i - queue chosen uniformly at random ; this is referred to as jiq - random .alternatively , a server can probe the length of i - queues and choose the one with the smallest number of jobs .this approach of using a constant number of choices for placement has been studied in many settings , in particular in the context where jobs choose one of a collection of parallel servers for service on arrival , and is referred to as sq .( the approach is sometimes referred to as the power of two choices , or balanced allocations .the variation where jobs arrive to parallel servers and choose one according to sq is often referred to as the `` supermarket model '' . ) in the context of jiq , this variation is referred to as jiq - sq . using sq to distribute idle servers among i - queues helps ensure that i - queues do not empty , avoiding the need to send incoming jobs to random servers instead of idle ones .in the equations that follow , we consider the evolution of i - queues and servers .recall is the numbers of i - queues , the number of servers , and .as done in , we consider the regime where is held constant and and are growing to infinity .we use to represent the number of i - queues with queued servers , and be the corresponding fraction of i - queues with queued servers . for and , we use to represent the number of servers with jobs that are in position in one of the i - queues , and similarly we use to represent the corresponding fraction of servers .finally , we use to represent the number of servers with jobs that are not currently in an i - queue , and similarly we use to represent the corresponding fraction of servers .  to begin , consider a short time interval , and consider the change in over that time interval .we use to denote  ] , where can include the value . for and , we find the first term , , represents the change in due to the completion of jobs at the servers , changing the number of jobs at the server .the second term is the probability that a new job arrives to an empty i - queue , and then is sent randomly to a server with either or jobs that is on its i - queue .the last term represents the change in due to an arrival that reduces the i - queue length at one of the i - queues holding a server that contributes to or .this equation simplifies in the fluid limit to   the case where the number of jobs at the server is 0 is similar : here the term corresponds to servers with one job and not on an i - queue finishing their job and then joining an i - queue with other servers .finally , we also need to handle servers that are enqueued . for , here the first terms correspond to job completions at servers with or jobs that are not on an i - queue ; the next terms correspond to an arrival at an empty i - queue that lands at a server not on an i - queue ; and the term corresponds to an arrival at an i - queue where the job at the front goes from having jobs to jobs .  similarly , for ,    an equilibrium distribution for the above system of equations satisfies that all the derivatives equal 0 .we show how to find the equilibrium distribution ; let us use and for the corresponding values in equilibrium .our process will be to show that all values can be written in terms of , and then that there is a unique value of that can be found numerically that satisfies equilibrium conditions .various equilibrium conditions arise from first principles and from the equations above .for example , in an equilibrium , , since the fraction of idle queues must be .equations ( [ eqn : qi1 ] ) and ( [ eqn : q01 ] ) yield in an equilibrium , since is the total fraction of servers that are not idle .it follows that .  the final key equation for determining the equilibrium distribution is the following : for and , to see this , consider a server with jobs and in queue at some time .the last time it chose an i - queue , suppose instead it had chosen an i - queue with length one larger than the one it actually chose , and then consider coupling all subsequent events except for interchanging the events ( arrivals , servers joining ) at these two i - queues . from this coupling , and from the fact that the ratio between the equilibrium probability of consecutive i - queue lengths is constant , we see that the relative probability that this server is in its i - queue instead of is in fact the ratio between the equilibrium probability of consecutive queue lengths , .we then have hence and   from equations ( [ eqn : s101 ] ) and ( [ eqn : si01 ] ) , we can now solve for the values .setting the derivative to 0 in equation ( [ eqn : s101 ] ) yields similarly , setting the derivative to 0 in equation ( [ eqn : si01 ] ) yields and inductively we can solve for all in terms of .equation ( [ eqn : s0j1 ] ) allows us to find in terms of by looking at when and setting the derivative to 0 : equation ( [ eqn : s0j1 ] ) then again allows us to find the remaining values of inductively , in terms of .we have tested these equations by doing a binary search to find the appropriate value of that leads to a solution where the sum of the values equals one , and have found it matches the results found in our empirical evaluation below in section [ sec : verify ] .we have not , to this point , been able to prove formally that there is unique fixed point equilibrium to these equations , although our testing of these equations thus far suggests that this is the case .we have also not yet found a way to express the equilibrium in a convenient form in terms of .   to test the accuracy of our equations , we ran a simulation of the actual queueing system . in our results below ( and throughout the paper ) , we started at an empty state and ran for 10000 units of time , with 10000 servers and 1000 dispatchers , so that .( we chose as this was the primary setting used in simulations in . )we focus on the average time in system for a job for convenience , although our simulations show more generally that our equations also track the load distribution of servers and i - queues accurately as well .we report the average time for simulations for jobs that complete after time 5000 and before time 10000 ; this gives the system time to reach approximate equilibrium , for suitable comparison .in reporting the average time in system , we take the average of 1000 trials ; we note the standard deviation over experiments is small .the results from our family of differential equations are derived from the average server load and little s law .we started the differential equations with all i - queues empty and all servers with one job initially for convenience .we used the euler method with a step size of ; experiments with other step sizes led to slightly different results for the average time in system but we found the differences were or less ; we therefore felt the step size was appropriate for presenting results .finally , we also compare with the results from .specifically , for the case of exponentially distributed service times and poisson arrivals , based on their assumptions they derive an analytical expression for the average time in system , given by   our results of table [ tab : table1 ] demonstrate that our fluid limit approach is significantly more accurate , particularly under higher loads , as one might expect given the assumptions use in . in these experiments ,the magnitude of our relative error is , in the worst example , about , while for the previous approximation the magnitude of the relative error ranges from roughly to ..the differential equation method provides accurate estimates of the average time in system for jiq - random at 10000 queues , with relative error of about 0.1% at and smaller relative error for lower rates .the results are much more accurate than the still quite good approximation derived in . [ cols="^,^,^,^ " , ]   table [ tab : table5 ] is not the entire story .when we look at the maximum load over our simulation , we also see the performance difference clearly , even with an early threshold . for sq , the maximum load at the end time over all of our simulations when is 10 , and when it is 6 . for jiq - random with an early threshold , the maximum load at the end time of our simulation when is always over 20 , and is sometimes over 30 ; when , it is 11 .  we emphasize that under larger values of , which may be suitable for real - world systems , these concerns are lessened , as higher arrival rates are needed for there to be large numbers of empty i - queues .we have derived fluid limit equations that describe the distributed join - idle - queue process under poisson arrivals and exponentially distributed service times , showing that these systems are amenable to asymptotic analysis .our methodology allows us to obtain very accurate estimates for reasonably sized finite systems . by examining variations ,we have shown the generality of this method , and in particular we have analyzed the compelling approach of serving i - queues while still processing one or more jobs . based on its reasonable performance under low loads and its significantly better performance in terms of both expectation and variance under higher loads ,we recommend strongly that builders of systems that want to use the distributed jiq approach examine the use of an early threshold .  on the theoretical side , for many of the jiq variations we have examined ,it is worthwhile to consider whether the asymptotic equilibrium distributions have a useful or pleasant form that could aid in rapidly calculating system performance or making comparisons among system .we leave this for future work .y. lu , q. xie , g. kliot , a. geller , j. larus , and a. greenberg .join - idle - queue : a novel load balancing algorithm for dynamically scalable web services ._ performance evaluation _ , 68(11):1056 - 1071 , 2011 .m. mitzenmacher , a. richa , and r. sitaraman .the power of two choices : a survey of techniques and results , in the handbook of randomized computing , edited by p. pardalos , s. rajasekaran , j. reif , and j. rolim .kluwer academic publishers , 2001 , pp .255 - 312 .n. d. vvedenskaya , r. l. dobrushin , and f. i. karpelevich .queueing system with selection of the shortest of two queues : an asymptotic approach ._ problems of information transmission _ , 32(1):15 - 27 , 1996 .

Abstract: in the context of load balancing , lu et al . introduced the distributed join - idle - queue algorithm , where a group of dispatchers distribute jobs to a cluster of parallel servers . each dispatcher maintains a queue of idle servers ; when a job arrives to a dispatcher , it sends it to a server on its queue , or to a random server if the queue is empty . in turn , when a server has no jobs , it requests to be placed on the idle queue of a randomly chosen dispatcher .   although this algorithm was shown to be quite effective , the original asymptotic analysis in makes simplifying assumptions that become increasingly inaccurate as the system load increases . further , the analysis does not naturally generalize to interesting variations , such as having a server request to be placed on the idle queue of a dispatcher before it has completed all jobs , which can be beneficial under high loads .   we provide a new asymptotic analysis of join - idle - queue systems based on mean field fluid limit methods , deriving families of differential equations that describe these systems . our analysis avoids previous simplifying assumptions , is empirically more accurate , and generalizes naturally to the variation described above , as well as other simple variations . our theoretical and empirical analyses shed further light on the performance of join - idle - queue , including potential performance pitfalls under high load .   = 1


device - to - device ( d2d ) communication among mobile users is drawing considerable attention for the development of next - generation wireless communication systems .the d2d communication functionality can enable various applications and services ( see , e.g. ) , such as proximity - based applications involving discovering and communicating with nearby devices ( e.g. , internet of things ) .such functionality can also enable higher data rates and system capacity by leveraging the underlying peer - to - peer wireless network that can be created via local communication among the users ( see , e.g. ) .moreover , incorporating caching capability into d2d communication networks have been shown to also significantly enhance the system throughput for applications that follow a popularity pattern , such as the on - demand video traffic for which a few dominant videos account for a large part of the traffic  .  however, considering the increasing density of mobile users in wireless networks , the problem of spectrum sharing and interference management inside d2d communication networks becomes of vital importance for the aforementioned applications and improvements .the main challenge for interference management in such networks is that neither fully coordinated synchronous cellular - type approaches that rely on advanced physical layer designs , nor fully distributed and asynchronous wifi - type mechanisms ( such as csma / ca ) are adequate .the downside of the first type of interference management mechanisms is that they need levels of centralization , coordination , and information at the mobile nodes that are difficult to accomplish in practice . on the other hand ,the problem with the second type of approaches is that their performance degrades significantly as the number of links grows .these issues have motivated a more recent approach that is based on a minimal level of coordination among the links which also maintains its promising performance for large numbers of links .this scheme , called flashlinq , is a distributed scheduling scheme which demonstrates considerable improvement over pure csma / ca . in a system of multiple source - destination pairs ( links ), this scheduling algorithm first orders the links according to a randomly selected priority list .then , starting from the higher - order links , each link is scheduled if it does not cause and does not receive `` much '' interference from the already scheduled links .the level of acceptable interference is determined based on the observed signal - to - interference ratio ( sir ) at all the previously scheduled links _ and _ also the current link .flashlinq has also been implemented and experimented in practice and shown to demonstrate promising performance compared to previous scheduling schemes .flashlinq scheduling can also be viewed as a refinement of the conventional independent set scheduling which is based on using a conflict graph to model the interference among the links ( see , e.g. and the protocol model in ) . in the independent set scheduling approach ,two links ( source - destination pairs ) are considered to be mutually non - interfering , hence able to transmit data at the same time , if the interference that they cause on each others destinations is below a certain threshold .the drawback of this scheme is that this threshold is set at a fixed value ( often at noise level ) which does not capture the effect of the number of links , their density inside the cell area , etc .more importantly , the scheme does not consider the signal - to - noise ratio ( snr ) level that each link itself can achieve and only takes the interference levels into account .flashlinq , however , overcomes this problem by comparing the direct signal power level that each link gets with its incoming interference power level .also , in the flashlinq scheduling algorithm , if a link does not cause / receive much interference to / from higher - priority links , but does not get a high direct signal power itself , it gets silent and `` yields '' such that lower - priority links have the opportunity to contribute more to the overall sum - throughput of the network .hence , both flashlinq and independent set scheduling approaches aim at finding subsets of links in which the interference among them is at a `` sufficiently '' low level , so that their simultaneous transmissions are not detrimental to each other .this gives rise to a natural question : what would be a theoretically - justified way of creating such subsets , and determining whether the interference among them is at a `` sufficiently '' low level ?  in this paper , we propose an answer to this question .we define a new concept of _ information - theoretic independent sets _ ( itis ) , which indicates the sets of links for which simultaneous communication and treating the interference from each other as noise is information - theoretically optimal ( to within a constant gap ) . in other words ,a subset of links forms an itis if by a simple scheme of using point - to - point gaussian codebooks with appropriate power levels at each transmitter and treating interference as noise at every receiver we can achieve the entire information - theoretic capacity region of that subset of links ( to within a constant gap ) .we use the recent optimality condition for treating interference as noise in to provide a description of itis based on the channel gains among the links in the network .in fact , as we will see later , a subset of links is defined to create an itis if for any link in the subset , the snr level is no less than the sum of its strongest incoming interference - to - noise ratio ( inr ) and its strongest outgoing inr ( all measured in db scale ) .it is important to note that this condition is quite different from that of flashlinq and independent set scheduling which only rely on thresholds on sir and inr values to identify the subsets of links with `` sufficiently '' low levels of interference .furthermore , we propose our new spectrum sharing mechanism , named _ information - theoretic link scheduling _( in short , itlinq ) , which schedules the links in an information - theoretic independent set to transmit data at the same time .we characterize the guaranteed fraction of the capacity region that itlinq is able to achieve in a specific network setting .in particular , we consider a set of source - destination pairs , where the source nodes are spread randomly and uniformly over a circular cell of fixed radius and each destination node is located within a distance of its corresponding source node . for the channel gains ,we only consider the path - loss model .in such a setting , we show that the criteria for defining information - theoretic independent sets transforms the network into a random geometric graph which enables us to characterize the fraction of capacity region that can be achieved by the itlinq scheme .in fact , depending on the value of , we identify three regimes in each of which itlinq can achieve a fraction of the capacity region within a gap of almost - surely :  * for , and . and defined on the set of positive integers , if and only if there exists a positive real number and a positive integer such that for all , .also , if and only if there exist two positive real numbers and and a positive integer such that for all , . ] * for , and * for , and .this shows a considerable improvement over the fraction of the capacity region that the conventional independent set scheduling can achieve , which is ( derived via numerical analysis ) .we will also show that the network model in which each destination gets associated with the closest source to itself is a subclass of the above model for any , and therefore we can asymptotically achieve a fraction of the capacity region in this case almost - surely .afterwards , we will focus on the challenge of distributed implementation of the itlinq scheme .we will develop a distributed spectrum sharing scheme based on itlinq , whose complexity is comparable to the flashlinq algorithm .the conditions that need to be satisfied for the sources and destinations in this scheme are based on the sufficient conditions for the optimality of treating interference as noise ( as mentioned in ) , hence providing a strong theoretical backbone for the algorithm .we will numerically evaluate the performance of our distributed scheme and compare it with flashlinq in an outdoor setting with 8 - 4096 links of random lengths spread uniformly at random in a square cell .we observe that the sum - rate achieved by the distributed itlinq scheme improves over that of flashlinq by more than 100% , while keeping the complexity basically at the same level .finally , we introduce a slight variation to the distributed itlinq scheme so as to consider fairness among the links in the network and show that our fair itlinq scheme achieves almost the same tail distribution for the link rates as in flashlinq , while demonstrating over 50% sum - rate gain .the rest of the paper is organized as follows . in section [ desc_itq ] ,we formally describe the notion of information - theoretic independent sets and the itlinq scheme and present a capacity analysis of this scheme . in section [ decent ] , we will propose a distributed version of the itlinq scheme . in section [ sim] , we numerically evaulate the performance of distributed itlinq and fair itlinq and compare them with flashlinq. we will conclude the paper in section [ conc ] .in this section , we introduce our scheduling scheme , which we call `` information - theoretic link scheduling '' ( in short , `` itlinq '' ) .we start by defining the notion of `` information - theoretic independent set '' ( in short , `` itis '' ) and then move forward to describe the itlinq scheme .afterwards , we will consider a specific network setting and in that setting , we will characterize the fraction of capacity region that itlinq is able to achieve to within a gap .we consider a wireless network composed of sources and destinations in which each source aims to communicate a message to its corresponding destination . all the links ( i.e. , source - destination pairs ) are considered to share the same spectrum , which gives rise to interference among all the transmissions .we assume that all the nodes ( i.e. , all the sources and the destinations ) know how many links exist in the network and they also agree on a specific ordering of the links , where by ordering we mean a labeling of the links from 1 to .furthermore , we assume that the nodes are synchronous ; i.e. , there exists a common clock among them .  the physical - layer model of the networkis considered to be the awgn model in which each source intends to send a message to its corresponding destination , and does so by encoding its message to a codeword of length and transmitting it within time slots .there is a power constraint of \leq p ] is a constant independent of .the achievable fraction of the capacity region expressed in theorem [ main_frac ] is only a _ lower bound _ on the fraction of the capacity region that itlinq is able to achieve , and therefore , itlinq can _ guarantee _ the achievability of this fraction of the capacity region ( to within a gap of bits ) .figure [ thm_comp ] illustrates the impact of the maximum source - destination distance decreasing rate on the fraction of the capacity region that can be achieved by itlinq ., the parameters are chosen such that the constant in theorem [ main_frac ] is equal to 1 . hence , any choice of the parameters for which is a valid choice . ] if the maximum source - destination distance is proportional to such that , then the itlinq scheme is capable of asymptotically achieving a fraction proportional to of the capacity region , within a vanishing gap . however, if the maximum source - destination distance scales as , then the achievable fraction of the capacity region decreases as which declines much slower than the previous case . in this case, the gap increases very slowly with respect to . finally , in the case that the maximum distance between each source and its corresponding destination scales faster than , we can achieve at least a _ constant _ fraction of the capacity region for asymptotically large number of links which is a considerable improvement , whereas the gap is increasing with the number of links .this matches the natural intuition that the closer the destinations are located to their corresponding sources , the more the signal - to - interference - plus - noise ratio and the higher the fraction of the capacity that can be achieved by the itlinq scheme .also , as a baseline , we have included the fraction of the capacity region that tdma and independent set scheduling can achieve , which is for both schemes .  as an immediate application of theorem [ main_frac ], one can consider the model in which all the source and the destination nodes are located uniformly and independently within a circular area of radius , and each destination gets associated with its _ closest _ source .for such a model , it is straightforward to show that itlinq can almost - surely achieve a fraction of the capacity region to within a gap of for any , when .   in order to characterize the fraction of the capacity region that itlinq is able to achieve and prove theorem [ main_frac ], we seek to find the minimum number of information - theoretic independent sets which cover all the links and we will then do time - sharing among these subsets . more precisely , if we denote the set of all the information - theoretic independent subsets of a network composed of source - destination pairs by , then we are interested in the minimum - cardinality subset of whose members cover all the links ; i.e. , their union is equal to the set of all the links .denote such a subset by and let .we will show that time - sharing among these information - theoretic independent sets can achieve the fractions of the capacity region mentioned in theorem [ main_frac ] .as the first step of the proof , we characterize the achievable fraction of the capacity region by the itlinq scheme and its gap with respect to the random variable in the following lemma .[ schfrac ] the itlinq scheme can achieve a fraction of the capacity region of a network composed of source - destination pairs to within a gap of .consider any rate tuple inside the capacity region of the network and consider any itis . from the result in , since tin is information - theoretically optimal in ( to within a constant gap ) , the rate tuple is achievable in the fraction of time which is allocated to , where   therefore , the rate achieved by any link in the network through the itlinq scheme , denoted by , can be lower bounded as where ( [ setcov ] ) follows from the fact that the subsets in cover all the links and that for every , we have .this completes the proof .  therefore , to find an achievable fraction of the capacity region by itlinq, we need to find an upper bound on , that is the minimum number of information - theoretic independent subsets which cover all of the links .one way to find such an upper bound is to restrict the tin - optimality condition in ( [ tin_opt ] ) . in other words ,we need to find another condition that implies condition ( [ tin_opt ] ) , but is more restricted and more tractable than ( [ tin_opt ] ) . imposing such a restricted sufficient conditionwill reduce the number of information - theoretic independent subsets , hence leading to an upper bound on . to this end , we present lemma [ two_indep ] . in the following ,we denote the distance between source and destination by and the distance between sources and by , .[ two_indep ] if in a network of source - destination pairs within the framework of the model in section [ cap ] , the distance between and satisfies , then   based on the model considered in section [ cap ] , we know that and .moreover , from the triangle inequality , we will have .similarly , we have .therefore , we can get and   combining ( [ snri ] ) and ( [ inrji ] ) , we will have and likewise , we can show that   combining ( [ inrsq ] ) with ( [ inrsqq ] ) yields . by symmetry, we will also have .this completes the proof .consequently , lemma [ two_indep ] implies that there exists a threshold distance of such that if the distance between two sources is greater than this threshold , the corresponding pair of links are considered to be information - theoretically independent ; i.e. , the interference they cause on each other is at a sufficiently low level that it is information - theoretically optimal to treat it as noise ( to within a constant gap ) .therefore , given an network of source - destination pairs with nodes spread as mentioned in the model in the beginning of section [ cap ] , we can build a corresponding undirected graph where is the set of vertices and if and only if ; i.e. , two nodes are connected together if and only if the distance between their sources is no larger than the threshold distance .we call the resultant graph the _ information - theoretic conflict graph _ of the original network .clearly , this graph is a random geometric graph .  to return to our original problem , note that we needed to find an upper bound on .the following lemma provides such an upper bound .[ chrom ] , where denotes the chromatic number .the chromatic number of is the smallest number of colors that can be assigned to all of the nodes of such that no two adjacent nodes have the same color .therefore , considering the subsets of the links which receive the same color , is the minimum number of subsets of the links which cover all the links and each of which consist of links whose sources have distance larger than . from lemma [ two_indep ] ,it is easy to show that if for three distinct links , all the pairwise source distances are larger than , then we will have that all the squared inr s within the subnetwork consisting of links are less than all the snr s . extending this argument, we can see that all the independent subsets of automatically satisfy the tin - optimality condition of ( [ tin_opt ] ) and hence are also information - theoretic independent subsets .therefore , , which denotes the minimum number of information - theoretic independent subsets that cover all the links , can be no more than , the chromatic number of .  thus the final step is to characterize the asymptotic distribution of . in this step, we will use parts ( i ) , ( ii ) and ( iv ) of theorem 1.1 in which we bring here for the sake of completeness . consider a positive integer and a norm on .suppose we have points in and a threshold distance , where .then , the random geometric graph is defined as a graph with vertex set in which vertices and are adjacent if and only if the . defining and to be equivalent to and , respectively, we have the following theorem .  [ th_rgg ] for the random geometric graph , the following hold .  * if for some fixed , then * if for all , then * if ( but still ) , then where is the unit ball in , is the `` maximum density '' of the distribution of nodes in and is the `` packing density '' , defined in .using theorem [ th_rgg ] , the following lemma characterizes the asymptotic behavior of the chromatic number of the information - theoretic conflict graph .[ chrbounds ] for the information - theoretic conflict graph , exhibits the following behavior as :  * if , then . * if , then . *if , then   since the information - theoretic conflict graph is a random geometric graph with threshold distance and the nodes are distributed in , we can make use of theorem [ th_rgg ] .we will have the following cases :  * if , then , and therefore we can use part ( iii ) of theorem [ th_rgg ] .note that the dominant term in is the first term , since .also , as mentioned in , for the case of euclidean norm in , we have and .also , since the distribution of the nodes is uniform on a circle of radius , we have .therefore , we can get . * if , then which converges to a constant asymptotically , since .this enables us to use part ( ii ) of theorem [ th_rgg ] , since for all .therefore , we have . * if , then , where .thus , we can make use of part ( i ) of theorem [ th_rgg ] to get   the proof of theorem [ main_frac ] then follows immediately from lemmas [ schfrac ] , [ chrom ] and [ chrbounds ] and also the fact that the continuous function preserves almost - sure convergence ( continuous mapping theorem ) .one of the most important phenomena in wireless networks is the concept of channel fading . even though fading seems to be a detrimental aspect of wireless networks, it can also be helpful if it is viewed in a careful way .probably the most well - known example for this is receive diversity at multi - antenna receivers , where we can make use of independently faded signals to combine them in the best way , leading to an improvement in the received snr .hence , it would be interesting to figure out how fading can affect the results we derived so far on the fraction of the capacity region that itlinq can achieve . in this section ,we focus on this problem , considering the same model for the spatial location of the nodes as in section [ cap ] with the difference that here , we consider the squared magnitude of the channel gain at distance to be where represents the rayleigh fade of the channel modeled as an exponential random variable with normalized mean of 1 .we consider a slow fading scenario ( i.e. , block fading ) , where the rate of change of the channel characteristics is much smaller than the rate of change of the transmitted signal .hence , the channel fade remains fixed during the transmission within each block of communication ( which corresponds to a scheduling phase of itlinq ) and changes i.i.d from one block to the next .the definition of information - theoretic independent set ( itis ) still remains the same as before , i.e. , within each block of communication with revisited channel gain values ( modeled as ) , a subset in which for any link condition ( [ itis_cond ] ) is satisfied is an itis in that block .obviously , introducing rayleigh fading into the channel model adds another source of randomness in the analysis of the fraction of the capacity region achieved by itlinq , which is due to the dependence of itis s on the random fade of the channels .however , we can still make use of lemma [ schfrac ] to characterize the fraction of the capacity region that itlinq can achieve in each block of communication when rayleigh fading is also included in the channel model . in this case , the faded interference may no longer be gaussian , but we can circumvent this issue due to the recent result in . in ,the authors show that in a multi - user network , gaussian noise is the worst - case additive noise in the sense that any rate tuple that can be achieved under the assumption of gaussian noise can also be achieved under non - gaussian additive noise of the same variance .therefore , by treating the aggregate ( non - gaussian ) noise plus faded interference at each destination as a gaussian noise , we achieve a lower bound on the achievable rate of itlinq . as a result ,lemma [ schfrac ] would still hold in a fading scenario , meaning that in each block of communication itlinq can achieve a fraction of the capacity region to within a gap of , where now , the minimum number of itis s whose union contains all the links , depends both on the spatial location of the links and the realization of the fading . characterizing the distribution of in the fading scenario ( even in the asymptote of ) is quite challenging , hence we will use numerical evaluation in the rest of this section to analyze the average fraction of the capacity that itlinq achieves in the fading scenario ( i.e. , ])in figure [ gaps ] .similar to the non - fading scenario of section [ cap ] , we note that , for the case of , the average gap again does not scale with the size of the network and always remains less than 1.2 bits , independent of the number of links .however , for the case of where a constant fraction of the capacity region can be achieved asymptotically , the gap increases with the number of links and theorem [ main_frac ] predicts that the increase is logarithmic with respect to .in this section , we present a distributed algorithm for putting the itlinq scheme into practice in real - world networks .the algorithm is inspired by the flashlinq distributed algorithm and its complexity is exactly at the same level as the flashlinq algorithm .however , as we will demonstrate through numerical analysis in section [ comp ] , it significantly outperforms flashlinq in a certain network scenario .as mentioned in section [ defs ] , we consider wireless networks consisting of source - destination pairs . in each execution of the algorithm , to address the issue of fairness among the links , we first permute the links randomly and reindex them from 1 to based on the realization of the random permutation , as also done in .this new indexing of the links corresponds to a priority order of the links : link has higher priority than link if , .then , link 1 is always scheduled to transmit at the current time frame and for the remaining links , each link is scheduled if it does not cause and receive `` too much '' interference to and from the higher priority links .the conditions for defining the level of `` too much '' interference for link are as follows , where is a design parameter :  * at , the following conditions must be satisfied : which imply that does not receive too much interference from higher - priority links . * at , the following conditions must be satisfied : which imply that does not cause too much interference at higher - priority links .  as it is clear ,there are two major differences here with respect to the flashlinq scheduling conditions : the first difference is that instead of considering the raw fraction , here we are considering an exponent for the snr term , which is completely inspired by the condition for the optimality of tin ( [ tin_opt ] ) .the second difference is that in condition ( [ itqdist2 ] ) , the outgoing interference of each link is compared to _its own _ snr rather than other links snr s .this is also inspired by the tin - optimality condition ( [ tin_opt ] ) .in fact , if the parameter is set to , then conditions ( [ itqdist1 ] ) and ( [ itqdist2 ] ) imply that the tin - optimality condition ( [ tin_opt ] ) is satisfied at link .this means that link can safely be added to the information - theoretic independent subset of higher priority links and get scheduled to transmit in the current time frame .this algorithm , therefore , seeks to find the largest possible _ distributed _ information - theoretic independent subset based on the priority ordering of the links .however , it is clear that selecting might be too pessimistic and restrictive , and may prevent some links which cause and receive low levels of interference from being scheduled. therefore , we will leave this variable as a design parameter , and as we will see in section [ comp ] , tuning this parameter can indeed improve the achievable sum - rate by this scheduling algorithm .the remaining question is : how can the sources and destinations check whether their pertinent conditions are satisfied ?this can be done by a simple signaling mechanism which is inspired by the flashlinq algorithm and is a two - phase process , in each of which we assume that each link uses its own frequency band and transmissions are interference - free :  * in the first phase , all the sources transmit signals at their full power .the destinations will receive their own desired signals and also all the interfering signals in separate frequency bands .then , the destinations estimate their received s and s and check if their desired conditions ( [ itqdist1 ] ) are satisfied .this phase is the same as that of flashlinq .* in the second phase , contrary to the `` inverse power echo '' mentioned in the flashlinq algorithm , the destinations also transmit signals at the same power level of the sources .similar to the first phase , in this phase all the sources can estimate the value of their desired s and s in order to verify the validity of condition ( [ itqdist2 ] ) .clearly , power control at the transmitters may lead to an improvement in the performance of the scheme .however , due to the complication in implementing power control among the links in a distributed way , we disregard it in our scheme and use full power at all the transmitters .see e.g. on power control algorithms in d2d underlaid cellular networks .as it is obvious , the complexity of our distributed signaling mechanism is completely comparable to that of the flashlinq algorithm .in this section , we numerically analyze the performance of distributed itlinq in two distinct settings .first , in section [ thm_sim ] we assess the performance of distributed itlinq under the model developed in section [ cap ] .furthermore , in section [ comp ] we compare the performance of distributed itlinq with flashlinq in a model similar to the one considered in .  for concreteness ,the algorithm used in this section for the performance evaluation of distributed itlinq is illustrated in pseudo - code format in algorithm [ dist_itlq ] . here, we assume that through multiple iterations of the training mechanism introduced in section [ decent ] , each link is aware of the values of its own snr and all its incoming and outgoing inr s and it is also aware of the active higher - order links .we assume that the knowledge of the active higher - order links is also available to each link in implementing the flashlinq scheme .* initialize * ; * for *  * if * at , =1 ; * endif * * if * at , =1 ; * endif * ; * end * * return *   for the implementation of itlinq distributively , we also consider a second tuning parameter which adds more flexibility to our scheme .this parameter can in general be tuned to optimize the performance of the algorithm in any network setting . for the results of this section , we will set to be equal to 25 db .algorithm [ dist_itlq ] returns a vector of length which specifies whether or not each link should be scheduled . in particular , for any , link is scheduled if and only if .   in this section ,we analyze the performance of distributed itlinq under the model of section [ cap ] . in section [ cap ]we characterized the fraction of the capacity region that itlinq can achieve ( to within a gap ) with full knowledge of all the channel gains for an asymptotically large number of links .one may wonder how well itlinq can perform under this model for _ finite _ number of links where each link only has a _ local _ knowledge of the channel gain values .this provides the motivation for studying distributed itlinq in this setting .  for the sake of numerical analysis in this section, we assume that the sources are distributed uniformly within a circle of radius and each destination is assumed to be uniformly distributed in a circle of radius around its corresponding source node where is chosen to be equal to . the squared channel gain at distance is taken to be equal to ( the constant is assumed to be normalized to 1 ) .the transmit power is taken to be 10 dbm and the additive white gaussian noise variance at the destination nodes is set to -110 dbm .we have evaluated the sum - rate achievable by distributed itlinq in this setting with the value of set to 0.5 and taking values 0.5 , 1 and 2 .the result is illustrated in figure [ thm_comp_sim ] .for the sake of comparison , we have also plotted the average sum - rate that can be achieved by time - sharing among the links .as the figure demonstrates , there is a huge sum - rate improvement by using distributed itlinq over time - sharing .moreover , while our theoretical analysis in theorem [ main_frac ] characterizes the fraction of the capacity region achievable by fully - centralized itlinq , what we observe in figure [ thm_comp_sim ] is that the network can still enjoy the significant sum - rate improvement that a distributed implementation of itlinq can provide .   in this section, we will illustrate the performance of our distributed algorithm and compare it with flashlinq through numerical analysis .we drop links randomly in a square .the length of each link , which is the distance between its corresponding source and destination , is taken to be a uniform random variable in the interval $ ] .as in , we use the carrier frequency of 2.4 ghz and a bandwidth of 5 mhz .the noise power spectral density is considered to be -184 dbm / hz . the transmit power is set to 20 dbm . moreover, the channel follows the los model in itu-1411 .in particular , if the base station antenna height is denoted by , the mobile station antenna height is denoted by and the transmission wavelength is denoted by , then the transmission loss ( in db ) at distance is taken to be equal to where denotes the breakpoint distance and denotes the basic transmission loss at the break point . as in , we assume all the antenna heights to be equal to , alongside with a log - normal shadowing with standard deviation of 10 db . the antenna gain per device is taken to be -2.5 db and the noise figure is assumed to be 7 db .figure [ comparison ] demonstrates the sum - rate achievable by the distributed itlinq scheme for different values of and its comparison to flashlinq .the implementation of flashlinq follows the same steps as in and in particular , the threshold values and are taken to be equal to 9 db .as the figure illustrates , tuning the parameter can lead to considerable gains over flashlinq . for the case of , in which conditions ( [ itqdist1 ] ) and ( [ itqdist2 ] ) are sufficient for the optimality of tin ( to within a constant gap ) , distributed itlinq exhibits over 28% gain compared to flashlinq for 4096 links .interestingly , setting results in more than 110% gain over flashlinq for 4096 links .however , as we increase to 1 , more and more links get scheduled which results in a degradation in the overall performance . as a baseline , the achievable sum - rate whenthere is no scheduling ( i.e. , all the links operate simultaneously ) is also plotted in figure [ comparison ] .moreover , in the same setting , we also study the cumulative distribution function ( cdf ) of the sum - rate in a network of 1024 links .the result is depicted in figure [ cdf ] .again , the same trend occurs in this plot , showing that distributed itlinq , especially for the value of , can result in considerable uniform gain compared to the sum - rate achievable by flashlinq .for instance , with 50% probability , the sum - rate achieved by flashlinq is less than 540 bits / sec / hz while with the same probability , the sum - rate achieved by distributed itlinq is less than 928 bits / sec / hz .another natural aspect of distributed scheduling schemes that is of considerable importance is the issue of fairness among the links . in particular, the scheduling scheme should take care of all links fairly , regardless of them being strong or weak .it can be seen that the distributed itlinq scheme favors strong links more than weak links . to highlight this issue ,a network with two links ab and cd is shown in figure [ ex_unfair ] .  in this figure, link ab is a low - snr link and link cd is a high - snr link .moreover , destination node b suffers from strong interference due to the source node c. to see why itlinq may be unfair in such a scenario regardless of the priority of the links , consider the following two cases :  * if link ab has a lower priority than link cd , link cd is first scheduled .then , destination b checks its scheduling condition and with a high probability may find that it is not satisfied ( since the interference from c is strong compared to the signal power received from a ) .this will prevent link ab from being scheduled . *if link ab has a higher priority than link cd , it will be scheduled first .then , since both destination node d is receiving a low amount of interference from a ( compared to the signal power from d ) and source node c is causing a low amount of interference at b ( compared to the signal power it delivers to d ) , link cd will also get scheduled and hurts the transmission of link ab .therefore , in both cases , the low - snr link ab will not get a high rate , if any .this motivates a modification of the distributed itlinq scheme to account for this issue . to this end, we present a _fair _ version of distributed itlinq as follows .inspired by the example in figure [ ex_unfair ] , in the fair itlinq algorithm , the high - snr links should get scheduled in a more restrictive way .this can be done by decreasing the parameters ( and ) in the scheduling condition for the outgoing interference of high - snr links . in general , ( and ) need to be a descending function of snr. however , one simple solution would be to choose a threshold such that if the snr of a link is higher than this threshold , and are altered to decreased values and .the pseudo - code for the fair itlinq scheme is presented in algorithm [ fair_itlinq ] .  * initialize * ; * for *  * if * at , =1 ; * endif * * if * * if * at , =1 ; * endif * * else * * if * at , =1 ; * endif * * endif * ; * end * * return *   to assess the performance of fair itlinq in terms of fairness , we have numerically evaluated the cdf of the link rates ( averaged over both priorities and locations ) for a network with 1024 links under the same model as the one mentioned in the beginning of this section .the threshold value for high - snr is chosen to be db and the modified parameters are set to and db . figure [ cdf_fair ] compares the cdf of the average link rate by distributed itlinq ( with ) , fair itlinq and flashlinq .as the figure illustrates , fair itlinq can improve the tail distribution of distributed itlinq and perform as well as flashlinq in terms of fairness .this certainly does not come for free and in fact , there is a trade - off between fairness and the achievable sum - rate . the sum - rate achieved by fair itlinqis compared with flashlinq in figure [ sumrate_fair ] .as the figure illustrates , for the case of 4096 links , the sum - rate gain of fair itlinq over flashlinq is more than 50% . for more information regarding the software implementation of itlinq ,the reader is referred to .in this paper , we introduced a new spectrum sharing scheme , called information - theoretic link scheduling ( itlinq ) , in order to manage the interference in wireless networks .the scheme relies on the recently - found optimality condition for treating interference as noise and at each time , schedules a subset of links in which treating interference as noise is information - theoretically optimal ( to within a constant gap ) .we presented a performance guarantee of the itlinq scheme by characterizing the fraction of the capacity region that it is able to achieve in a specific network setting .moreover , we developed a distributed way of implementing the itlinq scheme and showed , via numerical analysis , that it yields considerable gains over flashlinq , a similar recently - proposed scheduling algorithm .we also showed how to address the issue of fairness among the links in the network by introducing a fair version of the distributed itlinq scheme .there are multiple future directions to consider following this work .first , in this work we presented a lower bound on the fraction of the capacity region that itlinq is able to achieve .it would be interesting to see if there exists an upper bound on the achievable fraction of the capacity region by the itlinq scheme so that its performance can be characterized more precisely .this can be accomplished through new outer bounds on the capacity region of the interference channels .second , it is worthy to study the performance of the itlinq scheme under different models of d2d networks other than the ones presented in this work . in particular , models with time - varying topology that allow links to dynamically share the spectrum might be of interest .third , one can think about generalizing the itlinq scheme to multihop d2d networks .especially , the result in shows that coupling between interference management and relaying strategies can provide significant gains .hence , it would be worth figuring out the performance improvement that itlinq is able to guarantee in a multihop setting .fourth , it would be interesting to investigate the impact of more advanced interference management techniques , such as successive interference cancellation , on itlinq .for example , recent results in demonstrate that by a careful use of repetition coding at the transmitters and temporal interference neutralization at the receivers , one can achieve spectral efficiency gains that are considerably beyond the common interference avoidance approach .thus , an interesting future direction can be to characterize the impact of structured repetition coding and temporal interference neutralization on itlinq . finally ,as mentioned in section [ intro ] , flashlinq has been implemented in a practical testbed and its performance has been measured and evaluated .hence , it is interesting to also test the itlinq scheme and observe the improvements that it is able to yield in practice .n. naderializadeh and a. s. avestimehr , `` itlinq : a new approach for spectrum sharing , '' in proceedings of _ ieee international symposium on dynamic spectrum access networks ( dyspan ) _ , pp .327 - 333 , april 2014 .n. naderializadeh and a. s. avestimehr , `` itlinq : a new approach for spectrum sharing in device - to - device communication systems , '' to appear in proceedings of _ ieee international symposium on information theory _, june - july 2014 .k. doppler , m. rinne , c. wijting , c. ribeiro , and k. hugl , device - to - device communication as an underlay to lte - advanced networks , _ ieee communications magazine _ , vol .42 - 49 , dec . 2009 .g. fodor , e. dahlman , g. mildh , s. parkvall , n. reider , g. mikls , and z. turnyi , `` design aspects of network assisted device - to - device communications , '' _ ieee communications magazine _ , vol .170 - 177 , mar .m. zulhasnine , c. huang , and a. srinivasan , `` efficient resource allocation for device - to - device communication underlaying lte network , '' in proceedings of _ ieee 6th international conference on wireless and mobile computing , networking and communications ( wimob ) _ , pp .368 - 375 , oct . 2010 .q. duong and o. shin , `` distance - based interference coordination for device - to - device communications in cellular networks , '' in proceedings of _ fifth international conference on ubiquitous and future networks ( icufn ) _ , pp .776 - 779 , july 2013 .x. wu , s. tavildar , s. shakkottai , t. richardson , j. li , r. laroia , and a. jovicic , `` flashlinq : a synchronous distributed scheduler for peer - to - peer ad hoc networks , '' _ ieee / acm transactions on networking _ , vol .1215 - 1228 , aug .2013 .p. chaporkar , k. kar , and s. sarkar , `` throughput guarantees through maximal scheduling in wireless networks , '' in proceedings of _ 43rd annual allerton conference on communications , control , and computing _ ,x. lin , and n. b. shroff , `` the impact of imperfect scheduling on cross - layer rate control in multihop wireless networks , '' in proceedings of _ 24th annual joint conference of the ieee computer and communications societies ( infocom ) _ , 2005 .g. sharma , r. r. mazumdar , and n. b. shroff , `` maximum weighted matching with interference constraints , '' in proceedings of _ fourth annual ieee international conference on pervasive computing and communications workshops , ( percom ) _ , 2006 .g. sharma , r. r. mazumdar , and n. b. shroff , `` on the complexity of scheduling in wireless networks , '' in proceedings of _ the twelfth annual acm international conference on mobile computing and networking ( mobicom ) _, 2006 .v. r. cadambe and s. a. jafar , `` interference alignment and degrees of freedom of the -user interference channel , '' _ ieee transactions on information theory _ , vol .3425 - 3441 , aug .m. a. maddah - ali , a. s. motahari , and a. k. khandani , `` communication over mimo x channels : interference alignment , decomposition , and performance analysis , '' _ ieee transactions on information theory _54 , no . 8 , pp . 3457 - 3470 , aug .g. bresler , a. parekh , and d. n. c. tse , `` the approximate capacity of the many - to - one and one - to - many gaussian interference channels , '' _ ieee transactions on information theory _ ,9 , pp . 4566 - 4592 , sep .2010 .i. shomorony and a. s. avestimehr , `` degrees of freedom of two - hop wireless networks : everyone gets the entire cake , '' _ ieee transactions on information theory _ , vol . 60 , no .2417 - 2431 , may 2014 .  v. aggarwal , a. s. avestimehr , and a. sabharwal , `` on achieving local view capacity via maximal independent graph scheduling , '' _ ieee transactions on information theory _ ,5 , pp . 2711 - 2729 , may 2011 .[ ] navid naderializadeh received his b.s .degree in electrical engineering from sharif university of technology , tehran , iran , in 2011 and his m.s .degree in electrical and computer engineering from cornell university , ithaca , ny , in 2014 .he joined university of southern california , los angeles , ca , in 2014 , where he is pursuing his ph.d . in the department of electrical engineering .his research is focused on developing spectrum sharing mechanisms in wireless networks , especially in device - to - device ( d2d ) communication networks .navid obtained the first rank in the iranian nationwide university entrance exam in 2007 .he was also a recipient of jacobs scholarship in 2011 .[ ] amir salman avestimehr received the b.s .degree in electrical engineering from sharif university of technology , tehran , iran , in 2003 and the m.s . degree and ph.d .degree in electrical engineering and computer science , both from the university of california , berkeley , in 2005 and 2008 , respectively .he is currently an associate professor at the ee department of university of southern california , los angeles , ca .he was also a postdoctoral scholar at the center for the mathematics of information ( cmi ) at the california institute of technology , pasadena , in 2008 .his research interests include information theory , the theory of communications , and their applications .avestimehr has received a number of awards for research and teaching , including the communications society and information theory society joint paper award in 2013 , the presidential early career award for scientists and engineers ( pecase ) in 2011 , the michael tien 72 excellence in teaching award in 2012 , the young investigator program ( yip ) award from the u. s. air force office of scientific research in 2011 , the national science foundation career award in 2010 , and the david j. sakrison memorial prize in 2008 .he is currently an associate editor for the ieee transactions on information theory .he has also been a guest associate editor for the ieee transactions on information theory special issue on interference networks and general co - chair of the 2012 north america information theory summer school and the 2012 workshop on interference networks .

Abstract: we consider the problem of spectrum sharing in device - to - device communication systems . inspired by the recent optimality condition for treating interference as noise , we define a new concept of _ information - theoretic independent sets _ ( itis ) , which indicates the sets of links for which simultaneous communication and treating the interference from each other as noise is information - theoretically optimal ( to within a constant gap ) . based on this concept , we develop a new spectrum sharing mechanism , called _ information - theoretic link scheduling _ ( itlinq ) , which at each time schedules those links that form an itis . we first provide a performance guarantee for itlinq by characterizing the fraction of the capacity region that it can achieve in a network with sources and destinations located randomly within a fixed area . furthermore , we demonstrate how itlinq can be implemented in a distributed manner , using an initial 2-phase signaling mechanism which provides the required channel state information at all the links . through numerical analysis , we show that distributed itlinq can outperform similar state - of - the - art spectrum sharing mechanisms , such as flashlinq , by more than a 100% of sum - rate gain , while keeping the complexity at the same level . finally , we discuss a variation of the distributed itlinq scheme which can also guarantee fairness among the links in the network and numerically evaluate its performance .   device - to - device communication , interference management , distributed spectrum sharing


lsw experiments search for wisps via the process  . for a photon - counting detection scheme , the signal rate , ,is given by where is the rate of photons fed into the experiment , , the efficiency of the detector and the probability for photon - wisp and wisp - photon conversion , respectively , which are both proportional to the square of the photon - wisp coupling , .hence , the sensitivity on the coupling , , i.e. the expected upper limit on for the case that is realized in nature , scales with the detector parameters as with the count - rate sensitivity and the quantum efficiency of the detector .the count rate sensitivity is typically roughly proportional to the square root of the dark count rate , .hence , the sensitivity can be improved ( i.e. lowered ) by decreasing the dark count rate or increasing the quantum efficiency .  to compare different detectors ,the figure of merit , is used .thus , larger values of identify better detectors .the count rate sensitivity is taken to be the average upper limit of unified confidence intervals and is estimated using toy monte carlo simulations  .early lsw experiments used pmts for photo - detection  .recent lsw experiments used ccds and lasers in the visible spectrum  .future lsw experiments will use nir lasers  because optical elements are known to withstand high powers at these wavelengths . at nir wavelengths ,silicon based ccds have a much reduced quantum efficiency compared to the visible spectrum .therefore , other devices for photo - detection are sought .these detectors should have a quantum efficiency that is similar to the quantum efficiency of ccds in the optical and a dark count rate below that of ccds .additionally , it is desirable that these detectors can time - resolve single photons ( spd ) . a review of spds is given in ref . .  in sections [ seggern :sec : ccd ] to [ seggern : sec : pmt ] ( electron - multiplying ) ccds , qi spds from ref . and pmts are discussed , respectively , and the figures of merit are calculated .the results are compared in section [ seggern : sec : concl ] . to calculate the count rate sensitivity , , a confidence level of is assumed .ccds are currently the prime choice for scientific visual imaging with a wide range of devices to choose from .the imaging area of ccds is segmented into columns each consisting of a series of mis capacitors . during data taking, these capacitors are biased into deep - depletion .incident photons are absorbed in the semiconductor material and produce free charges which are stored by the capacitors .these charges are integrated during an exposure . at the end of an exposure ,the collected charges are transported to a read - out structure and digitized .hence , a ccd can not resolve single photons .in addition to the charges generated by incident photons , thermally generated free charges are produced and stored as well .these constitute the dark counts , which contributes to the overall noise .the process of read - out and digitization adds a second source of noise .hence , the total noise is given by where is the read - out noise , the exposure time and the production rate of dark counts .o0.37   the lsw experiment alps at desy used a commercially available , low noise ccd camera with sized pixels ( pixis ccd )  .a dark count rate below was achieved by liquid cooling of the ccd chip and the camera was equipped with low - noise read - out electronics ( )  .thus , the read - out noise is the larger contribution to the total noise for exposures shorter than .if the signal can be focussed to a single pixel , a data set of 20 one hour exposures yields a figure of merit for a quantum efficiency of which is typical in the visible spectrum .electron multiplying ccds ( emccds ) amplify the charge signal before read - out in an avalanche multiplication register  .this allows to neglect the read - out noise and , hence , short exposure times are possible .but at the same time , the quantum efficiency is effectively reduced by a factor of two due to the additional noise from the multiplication process  .the original quantum efficiency , i.e. without charge multiplication in an avalanche register , can be recovered by interpreting the read - out values in a binary fashion , i.e. photon detected yes / no  , where a photon is counted if the digitized signal is above a threshold , .hence , the analysis can be reduced to that of a counting experiment .if contamination by noise and loss of signal due to the threshold can be neglected , this yields assuming the same values as above ( , and of data ) .  for nir wavelengths ,the pixis ccd was found to have a reduced quantum efficiency of  .the figure of merit is reduced accordingly for the pixis ccd ( ) and emccd ( ) .ingaas based ccds exist , which have a much smaller band gap than silicon and , therefore , a much higher quantum efficiency ( ) than the silicon - based pixis ccd .but these devices also have a dark count rate , which is six orders of magnitude above that of the pixis ccd  .therefore , these specialized ccds are of no help when improving the detector part of lsw experiments .to maintain a low dark count rate and achieve a high quantum efficiency at nir wavelengths at the same time , sensors operated at cryogenic temperatures can be used .most of the devices listed in ref . ( cryogenic or not ) have however dark count rates much above that of the pixis ccd. only transition edge sensors ( tes ) were found to have low dark count rates below that of the pixis ccd  .tes are bolometric sensors which are operated at . combined with a proper coating, high quantum efficiencies of can be reached  .the dark count rate and quantum efficiency expected for alps - ii (  ) are assumed here as benchmark parameters  .the corresponding figure of merit for 20 hours of data is .the sensitive area of tes detectors and the pixels of a ccd are both of order .if the signal can not be focussed on such a small area , the pixels of a ccd can be binned .but , as discussed above , the integrated dark count rate increases at the same rate as the area of interest . accordingly , the figure of merit and the sensitivity on the coupling may worsen significantly . in this case ,pmts are a very good alternative although they have a limited quantum efficiency ( ) and a limited spectral range ( )  .the sensitive area of a  o0pt  [ cols="<,^,^,^",options="header " , ]   pmt consists of a photo - sensitive material with a low work function .incident photons produce free electrons which are directed to an electron multiplier by a focussing electrode .the high gain of the electron multiplier allows single photon detection . cooling the sensitive area reduces the dark count rate .for example , the ships helioscope uses a pmt with an active area of , which has a peak quantum efficiency of and a dark count rate of when cooled to  .this corresponds to .surprisingly , of all spds used in qi experiments , only tes detectors have a sufficiently low dark count rate to improve significantly over conventional ccds. the figures of merit of the detectors mentioned in the above sections are summarized in tab .[ seggern : tab : comparison ] . of the presented alternatives ,a tes is the best option . especially in the nir ,a tes is superior to a ccd because its quantum efficiency does not deteriorate for these wavelengths . in the visible regime ,ccds remain a viable option when only few resources are available for detector development . from the values listed in tab .[ seggern : tab : comparison ] , it seems that pmts are the worst option .their figure of merit is two orders of magnitude below that of ccds ( visible ) , which is caused mainly by their high dark count rate .however , considering their large sensitive area , pmts are the detector of choice if the signal can not be focussed very well .these findings are schematically summarized in fig .[ seggern : fig : findings ] .the study of the pixis ccd was kindly supported by g. wiedemann ( hamburger sternwarte ) .99 j. redondo and a. ringwald , contemp .phys .  * 52 * , 211 ( 2011 ) [ arxiv:1011.3741 [ hep - ph ] ] .g. ruoso , r. cameron , g. cantatore , a. c. melissinos , y. semertzidis , h. j. halama , d. m. lazarus and a. g. prodell _ et al ._ , z. phys .c * 56 * , 505 ( 1992 ) .k. ehret , m. frede , s. ghazaryan , m. hildebrandt , e .- a .knabbe , d. kracht , a. lindner and j. list _ et al ._ , phys .b * 689 * , 149 ( 2010 ) [ arxiv:1004.1313 [ hep - ex ] ] .p. pugnat _ et al ._ [ osqar collaboration ] , arxiv:1306.0443 [ hep - ex ] . + m. sulc [ osqar collaboration ] , _ contribution to this workshop_. r. bhre , b. dbrich , j. dreyling - eschweiler , s. ghazaryan , r. hodajerdi , d. horns , f. januschek and e. -a .knabbe _ et al . _ , jinst * 8 * , t09001 ( 2013 ) [ arxiv:1302.5647 [ physics.ins-det ] ] .+ b. dbrich [ for the alps - ii collaboration ] , _ contribution to this workshop _ [ arxiv:1309.3965 [ physics.ins-det ] ] .m. d. eisaman , j. fan , a. migdall and s. v. polyakov , rev .instrum . * 82 * , 7 ( 2011 ) [ doi:10.1063/1.3610677 ] .g. j. feldman and r. d. cousins , phys .d * 57 * , 3873 ( 1998 ) [ physics/9711021 [ physics.data-an ] ] .princeton instruments , http://www.princetoninstruments.com/uploads/princeton/documents/datasheets/pixis/princeton_instruments_pixis_1024_excelon_rev_n3_8.21.2012.pdf j. e. v. seggern , phd thesis , in preparation .s. madan , b. bhaumik and j. vasi , ieee ted * 30 * , 694 ( 1983 ) [ doi:10.1109/t - ed.1983.21191 ] . m. stanford and b. hadwen , ieee ted * 50 * 1227 ( 2003 ) [ doi:10.1109/ted.2003.813462 ] . o. daigle , c. carignan , j .- l .gach , c. guillaume s. lessard c .- a .fortin , s. blais - ouellette , publ .* 121 * 866 ( 2009 ) .princeton instruments , http://www.princetoninstruments.com/uploads/princeton/documents/technotes/ingaas_tech_note_princeton_instrumentsrevb0.pdf g. cantatore , _ contribution to this workshop_. + j. dreyling - eschweiler and d. horns , _ contribution to this workshop _ , [ arxiv:1309.5024 [ physics.ins-det ] ] .a. lita , a. miller , and s. nam , opt .express * 16 * 3032 ( 2008 ) .hamamatsu , `` photomultiplier tubes '' , edition 3a ( 2007 ) .et enterprises , 9215b , http://my.et-enterprises.com/pdf/9215b.pdf .m. schwarz , _ contribution to this workshop_.

Abstract: laboratory based searches for weakly - interacting slim particles ( wisps ) of the light - shining - through - a - wall type ( lsw ) use visible or near - infrared ( nir ) laser light . low - noise and highly efficient detectors are necessary to improve over previous experiments . these requirements overlap with the requirements for single - photon detectors ( spds ) for quantum information ( qi ) experiments . in this contribution , the sensitivity of several qi spds is compared to photo - multiplier tubes ( pmts ) and imaging charge - coupled devices ( ccds ) . it is found that only transition edge sensors ( tess ) are viable alternatives to ccds if the signal can be focussed to a few .


one of the promising directions of enhancement of the classical black scholes model is construction and research of diffusion models with volatility of risky asset governed by a stochastic process .empirical studies evidence in favor of the fact that the classical model with constant volatility is unable to capture important features of volatility observed in real financial markets .this drawback of the black scholes model has been widely investigated and to some extent eliminated by the extension of the theory in three directions : models with time - dependent deterministic volatility , models with state - dependent volatility , and models with stochastic volatility .the first and second of these categories may be viewed as intermediate between the classical model and third category , although equipping the market with certain constraints ( the most essential is the limiting time period under consideration ) allow less complex models to produce results of acceptable precision .  despite recent popularity of the stochastic volatility modification of the blackscholes theory , the range of models under consideration is quite narrow .one of the first models of such a type is presented in , where the authors assume the volatility of the price of risky asset to be governed by the square root of the geometric brownian motion .an expression for the price of european call option is derived under the following assumption : the volatility process is driven by a brownian motion independent of the brownian motion governing the price of risky asset . in , the authors choose the ornstein - uhlenbeck ( ou ) process to drive the volatility .the ou process is mean - reverting , and there is a strong evidence that the volatility in real financial markets has such a feature . under this assumption ,the authors of describe the distribution of the price of risky asset and apply it to derive an estimate of the price of european call option . as an alternative, there is an option to choose the cox ingersoll ross process to govern the volatility process .it is worth mentioning that although all cited works contain some significant results , they rely upon simplified models of real - world volatility process ( e.g. , ignoring the mean - reversion property ) .we remark that there is a vast amount of further investigations that consider more sophisticated and thus more realistic models .an extensive overview of these results is given in .nonnegativity is another desirable feature of the process modeling volatility .one of possible choices is to use the exponential function of the ou process ( see , and references therein ) .questions of existence of equivalent ( local ) martingale measures are investigated in different frameworks and different generality in . often , after specifying the model , the authors state that a risk - neutral measure exists and continue investigation in the risk - neutral world without defining the measure .a significant part of works ( including aforementioned ) use the fourier transform to derive an analytical representation of the price of european call option .a great deal of information about developments in application of the fourier transform to option pricing problems can be found in .our work investigates the market defined by a diffusion model with stochastic volatility being an arbitrary function governed by the ornstein uhlenbeck process . under generalsetting and quite mild assumptions , we prove that the market satisfies two distinct no - arbitrage properties for different classes of trading strategies .for the special case of uncorrelated wiener processes , we derive an analytical expression for the price of european call option .this paper is structured as follows : in section [ sec2 ] , we define a general model . in section [ sec3 ] , we present definitions and preliminary results necessary for further analysis . in section [ sec4 ] , we investigate matters of existence of equivalent ( local ) martingale measures and arbitrage properties of the general model . in section [ sec5 ] , we define a particular case of the general model and raise the problem of pricing european call option .section [ sec6 ] covers the derivation of an analytical expression for the option price .let be a complete probability space with filtration generated by correlated wiener processes , , .we consider the model of the market where one risky asset is traded , its price evolves according to the geometric brownian motion , and its volatility is driven by a stochastic process .more precisely , the market is described by the pair of stochastic differential equations      denote by and the deterministic initial values of the processes specified by eqs . .in sections [ sec2][sec4 ] , we impose the following assumptions :  * the wiener processes and are correlated with correlation coefficient  ] , is a martingale .therefore , , and by theorem [ nagcriteria ] we deduce that the market is .  in order to prove and properties of the market, we have proved the existence of one emm .however , we actually have a family of equivalent martingale measures in the market .the discounted price process is of the form  and thus is always a martingale given the square integrability of and any _ admissible _ choice of process .  let the market defined by ( [ model0])([model1 ] ) with assumptions ( a1)(a3 ) and additional condition ( [ sigma2x2 ] ) , and let the measure be such that   then .  since we have more than one equivalent martingale measure in the market , it is straightforward that the market is incomplete .each emm in the market is defined by the process associated with it . in the financial literature , the process is called the market price of volatility risk .  under the emm , the pair of processes has the following representation : where the processes are wiener processes w.r.t . according to the girsanov theorem , and and  are independent .  in the risk - neutral model ( [ modelq ] ) ,the volatility process is not the ornstein uhlenbeck process anymore .so , generally speaking , there is no analytic solution to the corresponding differential equation .therefore , we further consider a particular case of the general model , which is defined by a set of assumptions concerning the form and behavior of certain parameters of the model .let us define a modified set of assumptions :  ( b1 ) the wiener processes and are independent , that is , ;  ( b2 ) ( a2 ) ;  ( b3 ) ( a3 ) .assumption ( b1 ) simplifies the risk - neutral model to the following form : where are independent wiener processes w.r.t . .our purpose is to price a european call option in the model ( [ modelb ] ) .we limit further investigation to the valuation w.r.t . the minimal martingale measure .the emm in the market defined by the model ( [ modelb ] ) is minimal iff the process corresponding to is identically zero .suppose ,  ] , a.s .. then for and ,   the process is a local -martingale ; hence , is a local -martingale . by definition mmm .the converse statement of the theorem comes straightforward from the uniqueness of mmm .the solution of the differential equation defining the evolution of the price of asset has the following representation :   for a fixed trajectory of , the argument of the exponential function in the right - hand side of is a gaussian process , and , , has the log - normal distribution with , where , .this fact is crucial for the derivation of expression for the value of european call option .  the value of european call option at time 0 w.r.t .the mmm is defined by the general formula   we apply the telescopic property of mathematical expectation to transform the previous expression as follows :   the inner expectation is conditional on the path of , and therefore , it is actually the black scholes price for a model with deterministic time - dependent volatility . according to lemma 2.1 in , the inner expectation in has the following representation : where , and is the standard normal distribution function .notice that .the former notation may be viewed as the volatility averaged from the current moment to maturity , whereas the latter is the volatility averaged from the initial moment to the current one .notice that the inner conditional expectation is an increasing function of  ( see lemma 3.1 in ) , which is the type of behavior one may expect to be exhibited by the black scholes price of european call option .taking into account the form of inner integral , in order to derive an analytic expression for the price of an option , it is necessary to deal with expectation of . instead of trying to evaluate the integral analytically, it is possible to use the monte carlo method .from eqs .  we can see that in order to derive the formula for the option price , it is necessary to present the exact formula for the expectation of . in this section, we apply the inverse fourier transform after rearranging of the right - hand side of .we introduce the following deterministic functions , : we define the domains of each of these functions to guarantee the nonnegativity of the expressions under square root , that is , for , , and for , .[ ev0int ] suppose that the market is defined by the model ( [ modelb ] ) with assumptions ( b1)(b3 ) , is mmm , and is the price of european call option at time 0. then we have the following representations :         * the discriminant is a quadratic form w.r.t .there are two possibilities : * * .then for : , ; ; so * * .then for any , so * the discriminant is a quadratic form w.r.t .there are two possibilities : * * .then for , , ; .so * * .then for any .so    * for , * for , \notag & \quad + \mathbb{q}\bigl(\bar{\sigma}_0>\sigma_2(s)\bigr ) \bigr)\operatorname { e}^{-s^2/2}ds\\[-1.5pt ] \notag & \quad -\int_{-\infty}^{0 } \bigl(\mathbb{q}\bigl(\bar{\sigma}_0<\sigma_2(s)\bigr)-\mathbb { q}\bigl(\bar{\sigma}_0<\sigma_1(s)\bigr )\bigr)\operatorname{e}^{-s^2/2}ds\biggr ) \biggr)\\[-1.5pt ] \notag & \quad -k \biggl(\varphi(-l)+\frac{1}{\sqrt{2\pi } } \biggl(\int_{l}^{\infty}\bigl(\mathbb { q}\bigl(\bar{\sigma}_0<\sigma_4(s)\bigr)\\[-1.5pt ] \notag & \quad -\mathbb{q}\bigl(\bar{\sigma}_0<\sigma_3(s)\bigr)\bigr)\operatorname { e}^{-s^2/2}ds\\[-1.5pt ] & \quad -\int_{-\infty}^{-l } \bigl(\mathbb{q}\bigl(\bar{\sigma}_0<\sigma_3(s)\bigr)+\mathbb { q}\bigl(\bar{\sigma}_0>\sigma_4(s)\bigr )\bigr)\operatorname{e}^{-s^2/2}ds\biggr ) \biggr).\label{ev0_p2}\end{aligned}\ ] ]    assume that the probability density function of is piecewise continuous on .then due to the fourier inversion theorem , for almost all , the probabilities in the integrands in have the following representation : where is the characteristic function of .   [ optionpriceanrep ] suppose that the market is defined by the model ( [ modelb ] ) with assumptions ( b1)(b3 ) , is the mmm , and is the price at time 0 of european call option .let the probability density function of be piecewise continuous on .then we have the following representations :      indeed , in this case , is bounded as well , so the characteristic function admits the taylor series expansion around zero : where is the moment of the random variable , and .the moments of the random variable can be represented by applying the fact that the finite - dimensional distributions of the ornstein uhlenbeck process are gaussian vectors .bearing in mind that the covariance matrix of the process is nondegenerate and consists of the elements of the form we get the following representation for the moments of the random variable  : where   we have demonstrated that there is an analytic solution to the problem of pricing of european call option in the model . however ,the resulting formula is complicated and cumbersome . therefore , our further investigation will be aimed at comparison of numeric results produced by it with approximate calculations and possible simplifications .

Abstract: we consider the black scholes model of financial market modified to capture the stochastic nature of volatility observed at real financial markets . for volatility driven by the ornstein uhlenbeck process , we establish the existence of equivalent martingale measure in the market model . the option is priced with respect to the minimal martingale measure for the case of uncorrelated processes of volatility and asset price , and an analytic expression for the price of european call option is derived . we use the inverse fourier transform of a characteristic function and the gaussian property of the ornstein uhlenbeck process .   ./style / arxiv - vmsta.cfg  financial markets , stochastic volatility , ornstein uhlenbeck process , option pricing 91b24,91b25,91g20