Coverage probability in wireless networks with determinantal scheduling
CCoverage probability in wireless networks withdeterminantal scheduling
B. Błaszczyszyn, A. Brochard and H.P. Keeler
Abstract —We propose a new class of algorithms for randomlyscheduling network transmissions. The idea is to use (discrete) de-terminantal point processes (subsets) to randomly assign mediumaccess to various repulsive subsets of potential transmitters. Thisapproach can be seen as a natural extension of (spatial) Aloha,which schedules transmissions independently. Under a generalpath loss model and Rayleigh fading, we show that, similarly toAloha, they are also subject to elegant analysis of the coverageprobabilities and transmission attempts (also known as localdelay). This is mainly due to the explicit, determinantal form ofthe conditional (Palm) distribution and closed-form expressionsfor the Laplace functional of determinantal processes. Interest-ingly, the derived performance characteristics of the network areamenable to various optimizations of the scheduling parameters,which are determinantal kernels, allowing the use of techniquesdeveloped for statistical learning with determinantal processes.Well-established sampling algorithms for determinantal processescan be used to cope with implementation issues, which is isbeyond the scope of this paper, but it creates paths for furtherresearch.
I. I
NTRODUCTION
In wireless network research, an important challenge isscheduling the resources (essentially time, power and frequen-cies) to the network users, with the overall aim of allocatingthe service in some optimal manner. One way to achievesuch goals is to use (opportunistic) scheduling methods, wherethe algorithm makes decisions in real time, responding tothe changes of the network and traffic demands. There is alarge amount of research that proposes different schedulingtechniques, which range from simple heuristics to complexmathematical algorithms [1].In general terms, schedulers are simply algorithms thatwisely choose a subset of some underlying set with the aimof optimizing some utility function of that subset. Such subsetselection problems are known to be, in general, very hard tosolve computationally, as they are often NP-complete. Oneapproach to overcome this difficult is to leverage randomness.
A. Spatial Aloha
For wireless networks, the problem of scheduling or subsetselection is usually dictated by medium access control (MAC)protocols, with many being proposed over the years. A classicprotocol based on probability is the (spatial) Aloha scheme in
B. Błaszczyszyn is with
Inria/ENS , Paris, France; A. Brochard is with
Inria and
Huawei , Paris, France;
H.P. Keeler is with the
University of Melbourne and
ACEMS , Melbourne, Australia. This work was partly supported throughResearch Collaboration Agreement No. HF2016090005 between HuaweiTechnologies France and Inria on
Mathematical Modeling of 5G Ultra DenseWireless Networks . H.P.Keeler was supported by an Australian ResearchCouncil DECRA (ID: DE180100463). which network transmitters independently access the networkwith some probability p , where p is a fixed constant some-times called the medium access probability . Proposed in the1970s [2], this scheme has a long history and is particularlysuitable for Poisson network models, where the transmittersare scattered across the plane R according to a Poissonpoint process Φ = { X i } i with intensity λ > . Under theassumption of discrete time, at any time instant the transmittersaccessing the network will form another Poisson point process Φ p = { X i } i with intensity λp , due to the standard thinningresult of the Poisson point process.Baccelli, Błaszczyszyn and Singh [3] examined the case fortransmitters being allowed to have different p values dependingon the network configuration Φ . This ability to have different p value for each transmitter motivated the term adaptive Aloha . B. Determinantal point processes
Researchers often build random models of wireless net-works by using the Poisson point process to gain insightinto the coverage probability of a single user based on itssignal-to-interference-plus-noise ratio (SINR). But the Poissonpoint process does not exhibit repulsion between the points.To incorporate repulsion, researcher have developed networkmodels using specific types of determinantal point processes ,obtaining coverage results [4]–[7].Originally called fermion point processes , determinantalpoint processes were first proposed to model repulsive par-ticles, and they turn out to have useful mathematical proper-ties [8], with recent research showing that they are particularlyamenable to statistical inference methods [9]. For wirelessnetwork models, these point processes are defined typicallyon the plane R , but careful mathematical considerations andtechniques are needed in this setting. These point processesare more tractable when they are defined on discrete spaces,meaning there is a finite number of possible point locations,often reducing mathematical technicalities to linear algebra.In this setting, Kulesza and Taskar [10] used these pointprocesses to develop a (supervised) machine or statisticallearning framework for automatically choosing subsets, whichuses the concepts of point quality and diversity; see Section G,as well as other work by Kulesza and Taskar [11], [12]. C. Determinantal thinning
Recently Błaszczyszyn and Keeler [13] defined a new pointprocess existing on bounded regions of the plane R , whichcan be used as various types of network models; see Figure 1.They demonstrated that this new point process can be fitted a r X i v : . [ c s . N I] J un oisson processDeterminantal Poisson Fig. 1. A single realization ofa determinantally-thinned Poissonpoint process, where the remainingpoints exhibit repulsion.
13 4 567 89 2 10
Fig. 2. A single configuration oftransmitter-receiver pairs. to other repulsive point processes such as the Mat´ern hard-core types; also see [14] for code and examples. This newpoint process is obtained by using a discrete determinantalpoint process to define a tractable point process operationcalled determinantal thinning . Applied to the Poisson pointprocess, Błaszczyszyn and Keeler derived new results for theresulting point process, such as moment measures and Palmdistributions, as well as an accompanying statistical fittingmethod based on maximum likelihoods. They also discussedthe possibility of using determinantal point processes forscheduling wireless networks, with an emphasis on trainingthe determinantal point processes on pre-scheduled networks.This work initiated the current line of research.
D. Other work
Independently of our current work, Saha and Dhillon [15]recently tackled the scheduling problem with discrete determi-nantal point processes. Theirs and our current approach onlyoverlap slightly. They applied these point processes to thewireless link scheduling problem, with the aim of maximizingthe overall network rate, by reducing the problem down to oneof geometric programming, which is the subject of a recentreview [16, Chapter 3 and Appendix A].Conversely, our current work focuses on the special formof SINR problems, which allows for recasting the SINRexpression into a tractable (matrix) kernel for determinantalpoint process. We achieve this by using the elegant algebraicproperties of Palm distributions and Laplace functionals ofdeterminantal points processes.
E. Determinantal extends Aloha
Determinantal scheduling can be seen as a natural exten-sion of the Aloha medium access scheme, where the formerallows for repulsive patterns of simultaneous transmissions.In general, intuition says that the repulsive nature of determi-nantal point process should lead to better performance of thedeterminantal scheduler.
F. Current contributions
The current work contributes to the determinantal approachby treating the SINR coverage problem. Under the standardwireless signal propagation model, we show mathematicallythat the specific structure of the SINR problem combines ele-gantly with the special form of determinantal point processes, giving expressions for the coverage probabilities, which canthen be immediately evaluated numerically. To reproducethe results, we have uploaded the code; see the accompa-nying repositories for the code in either MATLAB [17] orPython [18].We believe that the determinantal scheduling is particularlyrelevant as a medium access scheme for device-to-devicenetworks, in which mobile users communicate directly witheach other under some central authority of the covering basestation, which is used to coordinate mutually dependent trans-missions. Well-established sampling procedures mean thatoperators can easily implement determinantal scheduling inwireless networks. We can also envision using a determinantalscheduling in other networks, such as a type of energy-savingsleep scheme [19], [20] or sentry selection [21] in wirelesssensor networks.II. D
ETERMINISTIC NETWORK MODEL WITH RANDOMPROPAGATION EFFECTS
We now consider a popular physical model of a wireless net-work with deterministic transmitters and receivers, and randompropagation effects. We assume a deterministic (that is, non-random) configuration of potential transmitters on the planewritten as φ = { x i } ni =1 ⊂ R . (The transmitters are on theplane, but the approach extends easily to higher dimensions.)For each transmitter x i ∈ φ , we consider a receiver locatedat y i , which in point process terminology is simply a mark ,resulting in a marked point process ˜ φ = { ( x i , y i ) } ni =1 , asillustrated in Figure 2. A. SINR
We write P x i ,y i to denote the received power at location y i of a signal emanating from a transmitter located at x i . For atransmitter x i ∈ φ , consider its SINR at location y i ∈ R withrespect to the active (concurrent) transmissions from nodesin a configuration ψ ! i ⊂ φ , to which transmitter x i does notbelong, meaning x i (cid:54)∈ ψ ! i . This SINR is given bySINR ( x i , y i ; ψ ! i ) = P x i ,y i W + (cid:80) x j ∈ ψ ! i P x j ,y i , (1)where W denotes the noise power.We now make some standard assumptions on the propaga-tion model. For a signal traversing a distance r , we assume itundergoes path loss according to the deterministic, continuous,non-negative function (cid:96) ( r ) , which is typically monotonic. Acommon choice is the power law (cid:96) ( r ) = ( κr ) − β , where κ > and β > , but our results hold for a more general path lossmodel, which is interesting in its own right.For any point y ∈ R , let independent random variable F x i ,y i describe the random fading of the signal propagatingfrom transmitter x i to location y i . Then the SINR expressionfor transmitter x i (cid:54)∈ ψ ! i becomesSINR ( x i , y i , ψ ! i ) = F x i ,y i (cid:96) ( x i − y i ) W + (cid:80) x j ∈ ψ ! i F x j ,y i (cid:96) ( x j − y i ) . (2) . Transmitter-and-receiver pairs We consider a transmitter-and-receiver pair ( x i , y i ) in thenetwork ˜ φ , so ( x i , y i ) ∈ ˜ φ , and a subset of active (interfering)transmitters ψ ! i that does not include x i . For τ ≥ , we write P ( SINR ( x i , y i , ψ ! i ) > τ ) (3)for the probability of a successful transmission from transmit-ter x i to its receiver y i given the set of active interferers ψ ! i .We call this probability the pair-coverage probability between x i and y i against ψ ! i . This is the probability that the randompropagation effects F x j ,y i allow one to achieve an SINR valuein equation (2) larger than some value τ .We now present a useful result for calculating these pair-coverage probabilities in a network with a path loss model (cid:96) and independent and identically-distributed (i.i.d.) Rayleighfading. Lemma II.1 (Elementary pair-coverage probabilities) . As-sume path loss model (cid:96) and i.i.d. exponential (with mean µ )distribution of all F x j ,y i variables. Then the pair-coverageprobabilities between transmitter x i and its receiver y i againstinterferers ψ ! i (cid:54)(cid:51) x i are given by P ( SINR ( x i , y i , ψ ! i ) > τ ) = w ( | x i − y i | ) (cid:89) z ∈ ψ ! i h ( | z − y i | , | x i − y i | ) , (4) where the functions h ( s, r ) := 1 τ (cid:96) ( s ) /(cid:96) ( r ) + 1 s, r ≥ , (5) w ( r ) := exp[ − ( τ /µ ) W/(cid:96) ( r )] r ≥ . (6)We present the complete proof in Section A, but we firstnote that Lemma II.1 is a variation of a result by Błaszczyszynand M¨uhlethaler [22, Lemma 3.1] who proved it for the caseof a power law path loss. Some additional factors appear inthe original result due to the Aloha medium access schemebeing assumed therein.To extend Lemma II.1, we consider all the active transmit-ters ψ := ψ ! i ∪ { x i } . The only source of randomness in ournetwork model is the random Rayleigh fading. But we canreplace the non-random active set ψ with a randomized subset Ψ ⊂ φ (resulting from a randomized scheduler or mediumaccess scheme) conditioned on x i ∈ Ψ . Then the producton the right-hand side of equation (4) takes the form of theprobability generating functional of the reduced Palm versionof Ψ , a fact which we will use in the next section.III. C OVERAGE PROBABILITIES IN A NETWORK WITHDETERMINANTAL MEDIUM ACCESS
A. Determinantal medium access generalizes Aloha
We define a determinantal (MAC) scheduler as a randomsubset Ψ ⊂ φ whose distribution is a determinantal point pro-cess on the state space φ with some (marginal determinantal)kernel K ; see Section D for more details and Section E forkernel examples. For a real symmetric matrix K indexed by the points of φ , having all its eigenvalues in the interval [0 , ,the distribution of Ψ is characterized by the finite-dimensionalprobabilities P (Ψ ⊇ ψ ) = det( K ψ ) , (7)where det denotes the determinant, and K ψ := [ K ] x i ,x j ∈ ψ denotes the restriction of K to the scheduled points ψ . Remark
III.1 . The determinantal (MAC) scheduler is a naturalextension of the adaptive Aloha described in Section I-A,where we consider it here for a finite network. We can see thisby taking the diagonal matrix kernel K = diag[( p ( x i )) x i ∈ φ ] ,for any function p ( x ) , which makes Ψ an independentBernoulli thinning of φ with probabilities p ( x i ) .We use a determinantal kernel K on the set of transmitters φ . We assume that the kernel K = K ( ˜ φ ) depends on theconfiguration of all potential transmitters and their receivers ˜ φ = { ( x i , y i ) } , but not on their fading variables. As describedin Section J, Ψ is a determinantal thinning of the transmitterconfiguration φ .To give the next results, we need some further notation. Fora function f ( x i ) , where x i ∈ φ , we write K { f } to refer to a(matrix) kernel defined as [ K { f } ] x i ,x j := (cid:112) − f ( x i )[ K ] x i ,x j (cid:113) − f ( x j ) , (8)where x i , x j ∈ φ . Furthermore, for a point x i ∈ φ , we definethe functions h x i ( x j ) := h ( | x j − y i | , | x i − y i | ) , x j ∈ φ \ { x i } , (9)and constants W x i ,y i := w ( | x i − y i | ) , (10)where functions h and w are given by expressions (5) and (6).Finally, Shirai and Takahashi [23, Theorem 1.7] proved thatthe reduced Palm distribution of a determinantal point processis the distribution of another determinantal point process witha modified kernel; also see Section H. More precisely, for x i ∈ φ , we write K ! x i to denote the kernel indexed by pointsof φ \ { x i } with entries [ K ! x i ] z i ,z j = [ K ] z i ,z j − [ K ] z i ,x i [ K ] z j ,x i [ K ] x i ,x i , z i , z j ∈ φ \ { x i } . (11)In other words, the conditional distribution of Ψ \ { x i } given the point x i ∈ Ψ is also the distribution of anotherdeterminantal point process Ψ ! x i with a modified kernel K ! x i given by equation (11). The (non-reduced) Palm distributionis obtained by appending the point x i to the point process Ψ ! x i , which is characterized by the kernel [ K x i ] z i ,z j , where z i , z j ∈ φ . For z i , z j ∈ φ \ { x i } , the kernel matrix entriescoincide with that of K ! x i , meaning [ K x i ] z i ,z j = [ K ! x i ] z i ,z j ,whereas the extra entries are one on the diagonal and zeroelsewhere.We now present our first main result for determinantalscheduler Ψ on the configuration φ with kernel K . Proposition III.1.
Assume a path loss model (cid:96) and i.i.d.exponential (with mean µ ) distribution of all fading variables.or a given transmitter x i ∈ φ and SINR threshold τ ≥ , theSINR distribution at receiver y i from transmitter x i , given x i is selected by the scheduler, is given by P ( SINR ( x i , y i , Ψ \ { x i } ) > τ | x i ∈ Ψ ) =det( I − K ! x i { h x i } ) W x i ,y i , (12) where I is an identity matrix. Furthermore, this probabilityalso incorporates the fading conditions and scheduler deci-sions. The proof requires re-writing equation (4) in the form of aLaplace functional and a conditioning argument related to thereduced Palm distribution of Ψ ; see Section B for the completeproof.For τ ≥ , we denote by P i ( τ ) the unconditional SINR(tail) distribution at receiver y i from transmitter x i ∈ φ , P i ( τ ) := P ( x i ∈ Ψ and SINR ( x i , y i , Ψ \{ x i } ) > τ ) , (13)which we call simply the coverage probability . We present anexpression for the coverage probability P i . Proposition III.2.
The coverage probability is given by P i ( τ ) = [ K ] x i ,x i det( I − K ! x i { h x i } ) W x i ,y i . (14) Proof:
Note that P { x i ∈ Ψ } = det( K { x i } ) = [ K ] x i ,x i .The result follows from Proposition III.1 by conditioning on x i ∈ Ψ .Note that the matrix ( I − K ! x i ) { h x i } on the right-hand-sideof (14) is indexed by the elements of φ \ { x i } . We extend itto the full configuration φ by taking [ K x i ( τ )] x j ,x k := [ I − K ! x i { h x i } ] x j ,x k if j, k (cid:54) = iW x i ,y i [ K ] x i ,x i if j = k = i, if j = i and k (cid:54) = i or k = i and j (cid:54) = i . (15)This new kernel allows us to express P i ( τ ) in a more compactway P i ( τ ) = det( K x i ( τ )) . (16)For a small number of pairs we can derive simple results. Example III.1.
Assume there are two transmitter pairs { x , y } and { x , y } with independent Rayleigh fading and apath loss model (cid:96) . The determinantal kernel, which is alwayssymmetric, takes the general form K = (cid:20) k k k k (cid:21) , (17)where the probabilities clearly ≤ k , k ≤ . (We needanother condition so that that the eigenvalues of K are alsobounded on the unit interval.) Then for the first transmitter-receiver pair, equation (16) quickly gives the coverage proba-bility P = k [1 − ( k − k /k )]¯ h w , where ¯ h := 1 − h x ( x ) and w := w ( | x − y | ) , whichare constants in terms of optimizing the network scheduling.Symmetry gives the coverage probability for the other pair. B. Transmission attempts (local delay)
The quantity P i ( τ ) is the probability that a single trans-mitter x i can successfully transmit a message to its receiver y i in a single transmission attempt. But this may not occurin the first attempt, motivating us to examine the number ofattempts needed for the first successful transmission, whichBaccelli and Błaszczyszyn [24] called local delay .For a transmitter x i ∈ φ , we will denote its local delayby L i = L ( x i , y i , φ ) . If we assume that at each attemptthe determinantal scheduler Ψ and fading variables F x,y aregenerated in an i.i.d. manner over successive time slots, theneach L i has a geometric distribution. Lemma III.3. L i has a geometric distribution with the mean E ( L i ) = 1 P i ( τ ) , (18) which implies P ( L i ≤ k ) = 1 − [1 − P i ( τ )] k . (19) Proof:
Given each attempt is independent of the previousone, L i is (by definition) a geometric variable with probability P i ( τ ) of success.The geometric distribution of L i allows for some straight-forward analysis. For example, if a transmitter x i ∈ φ makes k transmission attempts, then for the probability of having atleast one successful transmission to be greater than some value (cid:15) > the coverage probability needs to satisfy P i ( τ ) > − (1 − (cid:15) ) /k . (20)IV. T RANSMITTER - OR - RECEIVER NODES
We now consider a single set φ of network nodes. Weassume that if a node x i ∈ φ is not transmitting, then it can beacting as a receiver. Under a determinantal scheduling scheme Ψ , this means that a node x i ∈ φ is transmitting if x i ∈ Ψ ,and, conversely, a node x j ∈ φ can receive if x j (cid:54)∈ Ψ . We areinterested in the probability of transmitting from node x i ∈ φ to node x j ∈ φ , namely P i,j ( τ | x i ∈ Ψ , x j (cid:54)∈ Ψ):= P ( SINR ( x i , x j , Ψ \ { x i } ) > τ | x i ∈ Ψ , x j (cid:54)∈ Ψ) . (21)To present our next main result, we need the conceptof a two-fold (reduced) Palm distribution for two points x i , x j ∈ Ψ , which we simply obtain by applying recursivelyexpression (11) to point x i then x j (or vice versa, as orderdoes not matter). To be more specific, for a point x ∈ Ψ , wefirst write the reduced Palm distribution given by equation (11)as K ! x := K ! x ( K ) , where we interpret K x as a function (or anoperator) working on the matrix K . Then the kernel for thetwo-fold reduced Palm distribution is K !! x i ,x j = K ! x j ( K ! x i ( K )) , (22)where we use the (non-standard) superscript notation !! tohighlight that this is the Palm distribution reduced by two oints. Similarly, the equivalent equation for the (non-reduced)Palm distribution is K x i ,x j := K x j ( K x i ( K )) .In reality we will use a two-fold Palm distribution reducedby only one point , giving the “semi-reduced” Palm distributionkernel K ! x i ,x j = K x j ( K ! x i ( K )) . (23)We could use another approach, bypassing the need for thiskernel, but this is more suitable for practical implementations,as it allows an easy way to keep track of the indices of pointsand matrices when producing numerical results.We now present a result for calculating the probability P i,j ( τ | x i ∈ Ψ , x j (cid:54)∈ Ψ) ; see Section C for the proof. Proposition IV.1.
Assume path loss model (cid:96) and i.i.d. expo-nential (with mean µ ) distribution of all fading variables. Inthe network φ with determinantal scheduler Ψ , the probabilityof transmitting from node x i to node x j is given by P i,j ( τ | x i ∈ Ψ , x j (cid:54)∈ Ψ) = W x i ,x j (1 − [ K ! x i ] x j ,x j ) (24) × (cid:2) det( I − K ! x i { h x i } ) − [ K x i ] x j ,x j det( I − K ! x i ,x j { h x i } ) (cid:3) , where W x i ,x j = w ( | x i − x j | ) , function w is defined byequation (6) , K x i is a (non-reduced) Palm kernel, and K ! x i and K ! x i ,x j are respectively the reduced and semi-reduced (by x i ) Palm kernels.Remark IV.1 . Provided a singular path loss model (cid:96) ( r ) =( κr ) − β , if x j is a transmitter, then its signal power and, conse-quently, the interference are infinitely large at x j due to the sin-gularity in the path loss model, resulting in SINR ( x i , x j , , Ψ \{ x i } ) = 0 . Consequently, P ( SINR ( x i , x j , Ψ \ { x i } ) > τ | x i ∈ Ψ , x j ∈ Ψ) = 0 , which, in light of the proof of Proposi-tion IV.1, reduces equation (24) to P i,j ( τ | x i ∈ Ψ , x j (cid:54)∈ Ψ) = W x i ,x j (1 − [ K ! x i ] x j ,x j ) × (cid:2) det( I − K ! x i { h x i } ) (cid:3) , (25)removing the need for the two-fold (semi-reduced) Palm dis-tribution when (cid:96) ( r ) = ( κr ) − β . This observation underscoresthe unexpected effects that a singular path loss model can haveon SINR results.For τ ≥ , we denote by P i,j ( τ ) the unconditional SINR(tail) distribution at receiver x j from node x i ∈ φ , giving thecoverage probability P i,j ( τ ):= P ( x i ∈ Ψ , x j (cid:54)∈ Ψ and SINR ( x i , x j , Ψ \ { x i } ) > τ ) . (26) Proposition IV.2.
The coverage probability is given by P i,j ( τ ) := P ( x i ∈ Ψ , x j (cid:54)∈ Ψ) P i,j ( τ | x i ∈ Ψ , x j (cid:54)∈ Ψ) . (27) where P ( x i ∈ Ψ , x j (cid:54)∈ Ψ) = [ K ] x i ,x i − det( K { x i }∪{ x j } ) . (28) Proof:
The proof is similar to that of Proposition III.2, inaddition to P ( x i ∈ Ψ , x j ∈ Ψ) = det( K { x i }∪{ x j } ) , as givenby expression (7). V. C ODE
We have implemented all our mathematical results intoMATLAB and Python code, which is located in the respec-tive repositories [17] and [18]. We have also written thecorresponding network simulations. The mathematical resultsagree excellently with simulations, which reminds us thatdeterminantal point processes do not suffer from edge effects(induced by finite simulation windows). All mathematicaland simulation results were obtained on a standard desktopmachine, taking typically seconds to be executed.For a starting point, run the (self-contained) files
DemoDetPoisson.m or DemoDetPoisson.py to simu-late or sample a single determinantally-thinned Poisson pointprocess. The determinantal simulation is also performed by thefile funSimSimpleDPP.m/py , which requires the eigende-composition of a L kernel matrix; see Section F.The mathematical results for transmitter-and-receiverpair network, as described in Section II-B), areimplemented in the file ProbCovPairsDet.m/py ;also see funProbCovPairsDet.m/py . Themathematical results for transmitter-or-receiver network,as described in Section IV), are implementedin the file
ProbCovTXRXDet.m/py ; also see funProbCovTXRXDet.m/py . These files typicallyrequire other files located in the repositories.VI. D
ISCUSSION AND C ONCLUSION
In this work we have contributed to the line of researchthat shows that determinantal point processes are particularlysuitable as models of wireless networks. In addition to this, wehave proposed the determinantal scheduler, which is amenableto analysis, particularly due to its mathematical properties inrelation to the SINR.Our main new observation is the mathematical form of theSINR problem (with a general path loss model and Rayleighfading) allows one to treat SINR using results from deter-minantal point processes. Using this key insight, we derivednew (linear algebraic) expressions for the coverage probability(in terms of the SINR) for two wireless networks, in whichwe interpreted the new determinantal scheduler as a MACprotocol.But of course the determinantal scheduler and our anal-ysis can be applied to other types of wireless networks.For example, if one uses the determinantal scheduler as asleep scheme in a wireless sensor network, then one couldexamine the probability of a certain region containing a sensornode that wakes up and successfully measures and relaysinformation [20, Section 7].In addition to the numerically tractable SINR expressions,the existence of efficient sampling algorithms for (discrete)determinantal point processes is yet another motivation to usehem as schedulers. They are also have good properties forstatistically fitting data to models based them.Finally, practical research interest in determinantal pointprocess (on discrete spaces) has been strongly driven by the(supervised) machine or statistical learning work pioneeredby Kulesza and Taskar [10] who showed that these point pro-cesses are very suitable for tackling subset selection problems.But this work has not been in the context of wireless networks.We believe determinantal point processes show great promiseas scheduling (or MAC) schemes. Moreover, we believe thetools of (determinantal) machine learning can be used todesign schedulers and train them on pre-optimize wirelessnetworks, opening up future research avenues.A
PPENDIX
A. Proof of Proposition II.1
Given equation (2), we first consider the case of a singleinterferer z = ψ ! i existing, which gives P ( SINR ( x i , y i , ψ ! i ) > τ )= P { F ( x i ,y i ) (cid:96) ( | x i − y i | ) ≥ τ ( W + F ( z,y i ) (cid:96) ( | y i − z | ) } (29) = E [ e − ( τ/µ ) W/(cid:96) ( | x i − y i | ) − ( τ/µ ) F ( z,yi ) (cid:96) ( | z − y i | ) /(cid:96) ( | x i − y i | ) (cid:3) (30) = e − ( τ/µ ) W/(cid:96) ( | x i − y i | ) E (cid:2) e − ( τ/µ ) F ( z,yi ) (cid:96) ( | z − y i | ) /(cid:96) ( | x i − y i | ) (cid:3) , (31)where we use the assumption that the F variables are i.i.d.exponential random variables, and conditioning on F ( z,y i ) .(Recall the tail distribution P ( E > t ) = e − t/µ for anexponential random variable E with mean µ .) We see that thecoefficient term gives the function w defined by expression (6).Recalling the Laplace transform of an exponential variable E ( e − Et ) = 1 / (1 + µt ) , the expectation term further reducesto E (cid:2) e − ( τ/µ ) F ( z,yi ) (cid:96) ( | z − y i | ) /(cid:96) ( | x i − y i | ) (cid:3) = [1 + τ ( (cid:96) ( | z − y i | ) /(cid:96) ( | x i − y i | ))] − , (32)which yields function h defined by expression (5). The rest ofthe proof follows from induction and independence of the F variables. (cid:4) B. Proof of Proposition III.1
Conditioning on the event
Ψ = ψ ! i ∪ { x i } and using (4),we obtain P ( SINR ( x i , y i , ψ ! i ) > τ | Ψ = ψ ! i ∪ { x i } )= w ( | x i − y i | ) × exp − (cid:88) z ∈ ψ ! i log 1 /h ( | z − y i | , | x i − y i | ) . (33)Observe that P ( SINR ( x i , y i , Ψ \ { x i } ) > τ | x i ∈ Ψ )= E ! x i [ P ( SINR ( x i , y i , ψ ! i ) > τ | Ψ = ψ ! i )] , (34) where the expectation E ! x i is taken with respect to the reducedPalm distribution of the (determinantal) point process Ψ . Thenthe expression takes the form P ( SINR ( x i , y i , Ψ \ { x i } ) > τ | x i ∈ Ψ )= w ( | x i − y i | ) × E ! x i (cid:34) exp (cid:32) − (cid:88) z ∈ Ψ log 1 /h ( | z − y i | , | x i − y i | ) (cid:33)(cid:35) , (35)in which we see the Laplace functional of the point process Ψ under E ! x i , where we denote this point process by Ψ ! x i , giving P ( SINR ( x i , y i , Ψ \ { x i } ) > τ | x i ∈ Ψ )= w ( | x i − y i | ) L Ψ ! xi ( − log h ( | z − y i | , | x i − y i | )) . The Laplace functional of a determinantal point process (de-tailed in Section I) is expressed with a modified kernel givenby equation (49), yielding L Ψ ! xi ( − log h ( | z − y i | , | x i − y i | )) = det[ I − ¯ K ! x i ] , (36)where the kernel matrix ¯ K ! z , which is indexed by z i , z j ∈ φ \ { x i } , has the elements [ ¯ K ! x i ] z i ,z j :=[1 − h ( | z i − y i | , | x i − y i | )] / × [ K ! x i ] z i ,z j × [1 − h ( | z j − y i | , | x i − y i | )] / . (37)The kernel of Ψ ! x i is given by equation (11), completing theproof. (cid:4) C. Proof of Proposition IV.1
Define the events A := SINR ( x i , x j , Ψ \ { x i } ) > τ } , B := { x i ∈ Ψ } , and C := { x j (cid:54)∈ Ψ } , which implies P i,j = P ( A | B, C ) , as defined by expression (21). Writingthe complement of C as ¯ C = x j ∈ Ψ , then the total law ofprobability gives P ( A | B ) = P ( C | B ) P ( A | B, C ) + P ( ¯ C | B ) P ( A | B, ¯ C ) . (38)We know that − P ( C | B ) = P ( ¯ C | B ) = P x i ( x j ∈ ψ ) , whichis simply the diagonal element of (Palm) kernel K x i , so P ( ¯ C | B ) = [ K x i ] x j ,x j . (39)The left-hand side of equation (38) is P ( A | B ) = P ( SINR ( x i , x j , Ψ \ { x i } ) > τ | x i ∈ Ψ) , (40)which we obtain with Proposition III.1 by setting x j = y i . Wealso use Proposition III.1 for the remaining probability term P ( A | B, ¯ C )= P ( SINR ( x i , x j , Ψ \ { x i } ) > τ | x i , x j ∈ Ψ) (41) = E ! x i ,x j [ P ( SINR ( x i , x j , ψ ! i ) > τ | Ψ = ψ ! i )] , (42)where the expectation E ! x i ,x j is taken with respect to thereduced (only by x i ) two-fold Palm distribution of the (deter-minantal) point process Ψ , as given by equation(23). In otherwords, this term is calculated by using the reduced (only by i ) two-fold Palm distribution P ! x i ,x j conditioned on points x i and x j belonging to Ψ (instead of using the one-fold Palmdistribution). This distribution coincides with the distributionof a point process that is determinantal on φ \ ( { x i } ∪ { x j } ) with the kernel given by equation (11) applied recursivelyto first x i and then x j , which is then appended with x j ,completing the proof. (cid:4) D. Determinantal point processes
For a state space S with finite cardinality m , we considera real symmetric m × m matrix K indexed by the points of S , having all its eigenvalues in the interval [0 , . A discretepoint process is a determinantal point process Ψ defined withthe kernel K if for all configurations (or subsets) ψ ⊆ S , thefinite-dimensional probabilities are given by P (Ψ ⊇ ψ ) = det( K ψ ) , (43)where K ψ := [ K ] x i ,x j ∈ ψ denotes the restriction of K tothe entries indexed by the points in ψ , that is x i , x j ∈ ψ .Expression (43) implies that each diagonal entry K ii of matrix K is the probability of a location x i ∈ S being occupied bya point of the point process Ψ , meaning P ( x i ∈ Ψ) = K ii .Expression (43) also means that a determinantal point processis not uniquely defined by a single matrix K . E. Kernel examples
Usually one assumes the (marginal kernel) matrix K tobe positive semi-definite, recalling its eigenvalues need to bebounded between zero and one. (One can also characterize Ψ with a complex Hermitian kernel K .) Example A.1.
Consider a state space S consisting of threepoints, S = { x , x , x } ⊂ R , and define a determinantalpoint process with the kernel K = k k k k k k k k k , (44)where the diagonals are probabilities, meaning ≤ k , k , k ≤ . Then a further condition is needed to ensurethat the eigenvalues of K are also bounded on the unit interval.To populate the matrix K , one typically considers a kernelfunction K : S × S → R , such as the covariance functionsused to define Gaussian processes. Example A.2.
A typical example is the double exponential or Gaussian kernel function K ( x i , x j ) = Ce − ( | x i − x j | /σ ) , where x i , x j ∈ S , σ > , and C > is some suitable constantthat ensures the eigenvalues of K are properly bounded.It is not always obvious how to define new kernels, particu-larly when needing properly bounded eigenvalues, but we willsee in the Section F that they can be easily defined by usingthe formalism of L -ensembles. F. Kernels using L -ensembles The eigenvalues of the (non-negative definite) matrix L are real, non-negative, but need not be smaller than one. K has the same the eigenvectors as L , but its eigenvalues areequal to λ i / (1 + λ i ) , where λ i are the eigenvalues of L . Anydeterminantal kernel K having all eigenvalues strictly smallerthan one has the form (46) with L = ( I − K ) − − I . Moreover,we have for all ψ ⊂ S the probabilities P { Ψ = ψ } = det( L ψ )det ( L + I ) . (45)These types of determinantal point processes are called L -ensembles . They were originally studied in mathematicalphysics [25], but Kulesza and Taskar [10] applied them tothe machine learning problem of subset selection, building offtheir tractability for defining suitable kernels. G. Similarity and quality
A special class of determinantal point processes have ker-nels of the form K = L ( L + I ) − (46)for some real, symmetric, non-negative definite matrix L ; seeSection F for more details. We briefly recall the approachproposed by Kulesza and Taskar [10]. Consider a matrix L whose elements can be written as [ L ] x i ,x j = q x i [ S ] x i ,x j q x j , (47)for x i , x j ∈ φ , where q x is a positive function of x ∈ φ and S is a symmetric, positive semi-definite m × m matrix,where m = φ ) . These two terms are known as quality andthe similarity matrix . The quality q x measures the goodnessof point q x ∈ φ , while [ S ] x i ,x j gives a measure of similaritybetween points x i and x j . The larger the q x value, the morelikely there will be a point of the determinantal point process atlocation x , while the larger [ S ] x i ,x j value for two locations x i and x j , the less likely realizations will occur with two pointssimultaneously at both locations. H. Palm distributions
Determinantal point processes exhibit closure under Palmdistributions. Shirai and Takahasi [23, Theorem 1.7] provedthat for a general determinantal point processes, its Palm dis-tribution coincides with the probability distribution of anotherdeterminantal point process with a modified kernel.
Example A.3.
We recall the kernel of the three-point statespace given in Example A.1. For point x , the reduced Palmversion of the kernel is K ! x = (cid:20) k k k k (cid:21) − k (cid:20) k k k k k k (cid:21) (48)Shirai and Takahasi [23, Corollary 6.6] obtained similarPalm results for the n -fold Palm distribution when condition-ing on multiple points. Borodin and Rains [25, Proposition 1.2]independently derived Palm distributions for L -ensembles.But the connection between these two results has only beennvestigated recently [26, Section 5.7.4]. Błaszczyszyn andKeeler [13, Section B] presented another proof of the Borodinand Rains result. I. Laplace functional
For any non-negative function f , the Laplace functional ofthe detetermintal point process Ψ is given by L Ψ ( f ) : = E (cid:104) e − (cid:80) x ∈ Ψ f ( x ) (cid:105) = det[ I − K (cid:48) ] , (49)where the kernel matrix K (cid:48) has the elements [ K (cid:48) ] z i ,z j := [1 − e − f ( z i ) ] / [ K ] z i ,z j [1 − e − f ( z j ) ] / , (50)for all z i , z j ∈ φ . Shirai and Takahashi [27] proved this inthe general discrete case. But Błaszczyszyn and Keeler [13,Section A] presented a simpler, probabilistic proof, which usesthe finite state space assumption of the determinantal process. J. Determinantal thinning
To define determinantal thinning, we consider a (non-random) point pattern φ on some bounded region R ⊂ R .For the determinantal point process Ψ , we set the state spaceas the point pattern, so S = φ , resulting in a thinned pointpattern. More precisely, the points of the point pattern φ formthe state space of the finite determinantal point processes. Thepoint process Ψ is defined on a subset of the plane R , andthe points of the point pattern φ are dependently thinned suchthat there is repulsion among the points of Ψ . Example A.4.
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