Stability of the bipartite matching model

Stabilit y of the bipartite matc hing mo del Ana Bu ˇ si ´ c ∗ V arun Gupt a † Jean Mairesse ‡ Ma y 31, 2022 Abstract W e consider the bipartite matching mo del of customers and servers in tro duced by Caldentey , Kaplan, and W eiss (Adv. Appl. Probab., 2009). Customers and serv ers pla y symmetrical roles. There is a finite set C , resp. S , of customer, resp. serv er, classes. Time is discrete and at eac h time step, one customer and one server arriv e in the system according to a joint probabilit y measure µ on C × S , indep enden tly of the past. Also, at each time step, pairs of matche d customer and serv er, if they exist, depart from the system. Authorized matchings are giv en by a fixed bipartite graph ( C, S, E ⊂ C × S ). A matching p olicy is chosen, whic h decides ho w to matc h when there are sev eral p ossibilities. Customers/servers that cannot b e matc hed are stored in a buffer. The ev olution of the mo del can b e describ ed by a discrete time Mark ov chain. W e study its stability under v arious admissible matching p olicies includ- ing: ML (Matc h the Longest), MS (Matc h the Shortest), FIF O (matc h the oldest), priorities. There exist natural necessary conditions for stabilit y (independent of the matc hing policy) defining the maximal possible stabilit y region. F or some bipartite graphs, w e prov e that the stabilit y region is indeed maximal for an y admissible matching policy . F or the ML p olicy , we pro ve that the stability region is maximal for any bipartite graph. F or the MS and priority p olicies, w e exhibit a bipartite graph with a non-maximal stabilit y region. Keyw ords: Mark ovian queueing theory , stability , bipartite matching. Mathematics Sub ject Classification (MSC2010): 60J10, 60K25, 68M20, 05C21. Con ten ts 1 In tro duction 2 2 The bipartite matching mo del 3 2.1 State space description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Admissible matc hing p olicies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Necessary conditions for stability 7 3.1 Complexit y of verifying NCond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Connectivit y prop erties of the Mark ov chain 12 4.1 Stable s tructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Bac k to prop erty UTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Mo dels that are stable for all admissible p olicies 16 6 Priorities and MS are not alw ays stable 18 7 ML is alwa ys stable 22 ∗ INRIA/ENS, 23, av en ue d’Italie, CS 81321, 75214 Paris Cedex 13, F rance. E-mail: ana.busic@inria.fr . † Computer Science Department, Carnegie Mellon Universit y , Pittsburgh, P A, USA. E-mail: varun@cs.cmu.edu . ‡ LIAF A, CNRS et Univ. P aris 7, case 7014, 75205 P aris Cedex 13, F rance. E-mail: mairesse@liafa.jussieu.fr. 1 1 In tro duction In queueing theory , customers and serv ers play differen t roles. Customers arrive in the system, accum ulate in a buffer, get served by a serv er, and ev entually depart. Servers on the other hand alternate b etw een idle and busy p eriods but remain forev er in the system. Within this framework, man y v ariations and refinements are p ossible. F or instance, we may consider a mo del with multi-class customers and distinguishable serv ers. A customer of a given class c must c ho ose its server from a specified subset S ( c ) of the servers. And of course, the subsets S ( c ) ma y in tersect, see Figure 1. Figure 1: Queueing mo del of a call center. In this pap er, we consider a mo del with the same multi-class flav or, but in which, by contrast, customers and servers play completely identical roles. W e no w argue that this simple symmetry requiremen t leads in a natural and ineluctable wa y to the bip artite matching mo del . By symmetry , b oth customers and servers should arrive into the system and depart from it. More sp ecifically , up on completion of a service, both the customer and the serv er should depart sim ultaneously . T o mo del arriv als, w e hav e a priori more flexibilit y , but there is basically one non-trivial choice which is to assume that time is discrete and that customers and servers arriv e in pairs. Consider indeed the simplest possible mo del with contin uous-time arriv als: (i) there is only one class of customers and one class of serv ers; (ii) customers, resp. servers, arrive according to a P oisson process of rate λ , resp. µ ; (iii) services hav e duration 0. Let us describ e the state b y Z = X − Y , where X is the num b er of unmatched customers and Y the n umber of unmatc hed serv ers. The pro cess Z is a birth-and-death contin uous-time Marko v pro cess on Z with drift λ − µ . It is either transien t (if λ 6 = µ ) or null recurren t (if λ = µ ), but it is nev er p ositive recurren t. Let us switc h to discrete-time i.i.d. arriv als. At each time step, a batc h of customers and a batc h of serv ers arrive into the system. If the size of the batc hes are allo wed to b e different for customers and servers, then w e are back to the con tinuous-time situation, and ev en the simplest mo del is nev er positive recurrent. Therefore to get a non-trivial mo del, the natural assumption is that exactly one customer and one serv er arriv e in to the system at each time step. The resulting model is symmetric in another resp ect: b oth arriv als and departures occur in pairs. F or simplicit y , w e alwa ys assume that the service durations are null. So the mo del is specified b y: (i) the finite s et C of customer classes and the finite set S of serv er classes; (ii) the probabilit y la w µ on C × S for the arriv als in pairs; (iii) the bipartite graph ( C, S, E ⊂ C × S ) giving the p ossible matchings b etw een customers and serv ers (hence the possible departures in pairs); (iv) the matching p olicy to decide how to match when several choices are p ossible. W e consider so called admissible p olicies which dep end only on the curren t state of the system. Under these assumptions, the buffer con ten t ev olves as a discrete-time Mark ov chain. W e call this mo del the bip artite matching model. The bipartite matc hing mo del has b een in tro duced b y Caldentey , Kaplan, and W eiss [2], under an additional assumption of independence b et ween arriving customers and servers ( ∀ c, s, µ ( c, s ) = µ ( c, S ) µ ( C, s )), and for the FIF O policy . In their pap er, the authors mention sev eral possible domains of applications ranging from call centers to crossbar data switches. They also pro vide references to pap ers on related mo dels. W e refer the in terested reader to [2] for details. 2 In the bipartite matc hing model, there is an equal n umber of customers and servers at an y time. But the matc hing constraints ma y result in instability with unmatched customers and servers accum ulating. It turns out that proving stability , i.e. p ositiv e recurrence of the Mark o v chain, is highly non-trivial. Given a bipartite graph ( C, S, E ), there exist natural necessary conditions on µ for stabilit y to hold true. When these conditions are also sufficien t, w e say that the stabilit y region is maximal . Calden tey & al conjecture that any bipartite graph has a maximal stabilit y region for the FIF O p olicy [2, Conjecture 4.2]. They prov e the conjecture for some sp ecific mo dels (under the additional assumption of indep endence of customers and servers): (i) the N model defined by C = { 1 , 2 } , S = { 1 0 , 2 0 } , E = C × S − { (2 , 2 0 ) } , (ii) the W mo del, i.e. the matching mo del version of Figure 1; (iii) the NN model of Figure 2. In the first tw o cases, they are also able to compute explicitly the stationary distribution. F or the last case, the pro of of stabilit y is already in tricate. In the present pap er, w e consider the stability issue for v arious admissible matching p olicies: ML (Match the Longest), MS (Matc h the Shortest), FIF O (match the oldest), random (match uniformly), priorities. The irreducibility of the Marko v chain describing the mo del is not granted, and we first study this question in detail (Section 4). Then we obtain the following results: - sufficient conditions under which an y admissible p olicy is stable (Section 5); - for the NN mo del, the MS p olicy and some priorit y p olicies do not hav e a maximal stability region (Section 6); - for an y bipartite graph, the ML policy has a maximal stabilit y region (Section 7). W e do not kno w if the stabilit y region is alw ays maximal for the FIF O and random p olicies. Notations. Denote b y N = { 0 , 1 , 2 , . . . } the set of non-negative in tegers. Let A ∗ b e the free monoid generated by A . F or any w ord w ∈ A ∗ and an y B ⊂ A , set | w | B = # { i | w i ∈ B } , the n umber of o ccurrences in w of letters from B . F or B = { b } , w e shorten the notation to | w | b . F urthermore, for any w ∈ A ∗ , set [ w ] := ( | w | a ) a ∈ A (the commutativ e image of w ). 2 The bipartite matc hing model W e no w proceed to a more formal definition of the model. Definition 2.1. A bipartite matching structure is a quadruple ( C, S , E , F ) wher e • C is the non-empty and finite set of customer typ es; • S is the non-empty and finite set of server typ es; • E ⊂ C × S is the set of p ossible matchings; • F ⊂ C × S is the set of p ossible arrivals. The bip artite gr aph ( C , S, E ) is c al le d the matc hing graph . It is assume d to b e c onne cte d. The bip artite gr aph ( C , S, F ) is c al le d the arriv al graph . It is assume d to have no isolate d vertic es. The tw o assumptions in Def. 2.1 are made without loss of generalit y , see Remarks 1 and 2. In Figure 2 we giv e an example of a matching graph with 3 customer and 3 server t yp es, called the “NN graph” in the follo wing. 1 3 2 2’ 1’ 3’ C S Figure 2: NN graph. 3 Customers and serv ers pla y symmetrical roles in the mo del. Also E and F play dual roles. The graph ( C , S, E ) defines the pairs that ma y depart from the system, while the graph ( C , S, F ) defines the pairs that may arrive in to the system. Definition 2.2. A bipartite matching mo del is a triple [( C , S, E , F ) , µ, Pol ] , wher e • ( C , S, E , F ) is a bip artite matching structur e; • µ is a pr ob ability me asur e on C × S satisfying supp ( µ ) = F , supp ( µ C ) = C, supp ( µ S ) = S , (1) wher e µ C and µ S ar e the C and S mar ginals of µ . • Pol is an admissible matching p olicy (to b e define d in § 2.2). Observ e that we can simplify the notation to [( C, S, E ) , µ, Pol ]. W e say that the mo del [( C, S, E ) , µ, Pol ] is asso ciate d with the structure ( C, S, E , F ). Remark 1. F or (1) to hav e solutions, ( C , S, F ) m ust b e without isolated vertices, the assumption made in Definition 2.1. This is not a real restriction: if it is not satisfied, w e can consider a new mo del without suc h customer or serv er classes. A realization of the mo del is as follows. Consider an i.i.d. sequence of random v ariables of law µ , represen ting the arriv al stream of pairs of customer/server. A state of the buffer consists of an e qual num b er of customers and servers with no p ossible matchings b etw een the classes. Up on arriv al of a new ordered pair ( c, s ), tw o situations may o ccur: if neither c nor s match with the serv ers/customers already present in the buffer, then c and s are simply added to the buffer; if c , resp. s , can be matched then it departs the buffer with its match. If several matc hings are p ossible for c , resp. s, then it is the role of the matching policy to select one. An admissible p olicy selects according to the curren t state of the buffer (and not according to the whole history of the buffer conten ts, for instance). The resulting evolution of the buffer is describ ed by a discrete-time Mark ov chain. 2.1 State space description Dep ending on the matc hing policy , we consider either a commutativ e (e.g. for Random) or a non-comm utative (e.g. for FIF O) state space description. The different p olicies considered in the pap er will be formally defined in § 2.2. Let us c ho ose a matching graph ( C , S, E ). W e introduce the follo wing conv enien t notations: C ( s ) is the set of customer classes that can b e matched with an s -server; S ( c ) is the set of serv er classes that can be matc hed with a c -customer: S ( c ) = { s ∈ S : ( c, s ) ∈ E } , C ( s ) = { c ∈ C : ( c, s ) ∈ E } . F or an y subsets A ⊂ C , and B ⊂ S , w e define S ( A ) = ∪ c ∈ A S ( c ) , C ( B ) = ∪ s ∈ B C ( s ) . Comm utative state space. A state of the system is giv en by ( x, y ), x = ( x c ) c ∈ C and y = ( y s ) s ∈ S , where x c denotes the num b er of customers of type c and y s the num b er of servers of type s . The c ommutative state sp ac e is: E = n ( x, y ) ∈ N C × N S : X c ∈ C x c = X s ∈ S y s ; ∀ ( c, s ) ∈ E , x c y s = 0 o . (2) Non-comm utative state space. A state of the system is given b y tw o finite words of the same size k ≥ 0, resp ectiv ely on the alphab ets C and S , describing unmatched customers and serv ers. The non-c ommutative state sp ac e is: E = n ( u, v ) ∈ ∪ k ≥ 0 ( C k × S k ) : ([ u ] , [ v ]) b elongs to (2) o . (3) 4 F acet. Both the comm utativ e and the non-comm utative state space can b e decomp osed into facets, defined only b y the non-zero classes. Definition 2.3. A facet is an or der e d p air ( U, V ) such that: U ⊂ C , V ⊂ S and U × V ⊂ ( C × S − E ) . The zero-facet is the fac et ( ∅ , ∅ ) , we denote it shortly by ∅ . F or a facet F = ( U, V ), define: C • ( F ) = U, S • ( F ) = V , C } ( F ) = C ( V ) , S } ( F ) = S ( U ) , C ◦ ( F ) = C − ( C • ( F ) ∪ C } ( F )) , S ◦ ( F ) = S − ( S • ( F ) ∪ S } ( F )) . W e alleviate the notations to C • , S • , C } , . . . , when there is no p ossible confusion. The symbol • stands for the non-zero classes, the sym b ol } for the classes that are forced to be at zero (since they are matched with non-zero classes), and the symbol ◦ for the classes that happ en to b e at zero. The following notion will pla y an imp ortan t role later on. Definition 2.4. A fac et F is c al le d satur ate d if C ◦ ( F ) = ∅ or S ◦ ( F ) = ∅ . In Figure 3, the facet on the left is non-saturated, while the one on the right is saturated. 1 3 2 2’ 1’ 3’ 1 3 2 2’ 1’ 3’ Figure 3: NN graph: facets ( { 3 } , { 3 0 } ) and ( { 2 } , { 3 0 } ). Graphical conv en tion. A facet F can be represen ted graphically by coloring the no des of the bipartite graph according to the abov e con v ention (see Figure 3 for an illustration): - no des in C • ( F ) and S • ( F ) are represen ted as filled circles; - no des in C } ( F ) and S } ( F ) are represen ted as double circles; - no des in C ◦ ( F ) and S ◦ ( F ) are represen ted as simple circles. In Figure 4, w e ha ve represented the facets of the NN graph. The more complex case of the NNN graph will be giv en in Section 5, Figure 13. 0 {2},{3’} {3},{3’} {3},{2’} {2,3},{3’} {3},{2’,3’} {1},{1’} Figure 4: F acets for the NN graph. Algorithm 1 takes as input a matching graph and returns as output the set of facets. The termination and correctness of the algorithm are easily prov ed. 5 Algorithm 1: Computation of the facets Data : A bipartite graph G = ( C, S, E ). Result : F ac ets - set of all facets of G . b egin F ac ets ← ∅ ; New ← ∅ ; foreac h ( i, j ) ∈ C × S − E do New ← New ∪ { ( { i } , { j } ) } ; while New 6 = ∅ do F ac ets ← F ac ets ∪ New ; Old ← New ; New ← ∅ ; forall the H , K ∈ Old such that H 6 = K do if C • ( H ) = C • ( K ) or S • ( H ) = S • ( K ) then Z ← ( C • ( H ) ∪ C • ( K ) , S • ( H ) ∪ S • ( K )); New ← New ∪ { Z } ; F ac ets ← F ac ets ∪ {∅} ; return F ac ets ; 2.2 Admissible matching p olicies Informally , a matching p olicy is admissible if: - only the current state of the buffer is tak en in to accoun t; - priority is given to customers/servers that are already presen t in the buffer: if the state is ( u, v ) and the new arriv al is ( c, s ) ∈ E , then c and s are matc hed together iff there are no serv ers from S ( c ) in v and no customers from C ( s ) in u . It results from the first point that an admissible matching p olicy can b e describ ed as a mapping  : E × ( C × S ) → E which returns the new state of the system after an arriv al. The second p oin t is called the buffer-first assumption. It is not a real restriction: a matc hing p olicy that alwa ys giv es priority to new arriv als can b e seen as a sp ecial case of the ab o ve with an arriv al probability µ such that µ ( E ) = 0. W e now define admissible p olicies formally , distinguishing b et ween the non-comm utative and comm utative state spaces. F or a word w ∈ A k and i ∈ { 1 , . . . , k } , w e denote by w [ i ] := w 1 . . . w i − 1 w i +1 . . . w k the subw ord of w obtained by deleting w i . Definition 2.5 (Non-comm utative case) . A matching p olicy is admissible if ther e ar e functions Φ and Ψ such that: ( u, v )  ( c, s ) =            ( uc, v s ) , if | u | C ( s ) = 0 , | v | S ( c ) = 0 , ( c, s ) 6∈ E ( u, v ) , if | u | C ( s ) = 0 , | v | S ( c ) = 0 , ( c, s ) ∈ E ( u [Φ( u,s )] , v [Ψ( v ,c )] ) , if | u | C ( s ) 6 = 0 , | v | S ( c ) 6 = 0 ( u [Φ( u,s )] c, v ) , if | u | C ( s ) 6 = 0 , | v | S ( c ) = 0 ( u, v [Ψ( v ,c )] s ) , if | u | C ( s ) = 0 , | v | S ( c ) 6 = 0 The FIF O and LIFO policies are admissible matching policies with functions Φ and Ψ as follo ws: • FIFO : Φ( u, s ) = arg min { u k ∈ C ( s ) } , Ψ( v , c ) = arg min { v k ∈ S ( c ) } . • LIFO : Φ( u, s ) = arg max { u k ∈ C ( s ) } , Ψ( v , c ) = arg max { v k ∈ S ( c ) } . F or c ∈ C , let e c ∈ N C b e defined by ( e c ) c = 1 and ( e c ) d = 0 , d 6 = c . F or s ∈ S , let e s b e defined accordingly . 6 Definition 2.6 (Comm utative case) . A matching p olicy is admissible if ther e ar e functions Φ and Ψ such that: ( x, y )  ( c, s ) =            ( x + e c , y + e s ) , if x C ( s ) = 0 , y S ( c ) = 0 , ( c, s ) 6∈ E ( x, y ) , if x C ( s ) = 0 , y S ( c ) = 0 , ( c, s ) ∈ E ( x − e Φ( x,s ) , y − e Ψ( y ,c ) ) , if x C ( s ) 6 = 0 , y S ( c ) 6 = 0 ( x − e Φ( x,s ) + e c , y ) , if x C ( s ) 6 = 0 , y S ( c ) = 0 ( x, y − e Ψ( y ,c ) + e s ) , if x C ( s ) = 0 , y S ( c ) 6 = 0 The follo wing commutativ e matching p olicies are admissible (for RANDOM, ML, and MS p olicies Φ( u, s ) and Ψ( v , c ) are random v ariables): • PR (Priorities). F or each customer type c ∈ C , w e define a priority function α c : S ( c ) → { 1 , . . . , | S ( c ) |} . Similarly , for each serv er t yp e s ∈ S , w e define β s : C ( s ) → { 1 , . . . , | C ( s ) |} . In the case of several matc hing options, a customer/server is matched with the server/customer that has the highest priorit y (greatest v alue of the priorit y function). It is conv enient to spec- ify the priorities b y t w o | C | × | S | matrices A and B defined b y: A cs =  α c ( s ) , ( c, s ) ∈ E 0 , otherwise B cs =  β s ( c ) , ( c, s ) ∈ E 0 , otherwise . Then Φ( x, s ) = arg max { β s ( c ) : c ∈ C ( s ) , x c > 0 } and Ψ( y , c ) = arg max { α c ( s ) : s ∈ S ( c ) , y s > 0 } . • RANDOM : Φ( x, s ), resp. Ψ( y , c ), is a random v ariable v alued in C ( s ), resp. S ( c ), and distributed as  x i / P j ∈ C ( s ) x j  i ∈ C ( s ) , resp.  y i / P j ∈ S ( c ) y j  i ∈ S ( c ) . Intuitiv ely , the match is chosen uniformly among all p ossible ones. • ML : Φ( x, s ), resp. Ψ( y , c ), is a random v ariable uniformly distributed on arg max { x i : i ∈ C ( s ) } , resp. arg max { y i : i ∈ S ( c ) } . • MS : Φ( x, s ), resp. Ψ( y , c ), is a random v ariable uniformly distributed on arg min { x i > 0 : i ∈ C ( s ) } , resp. arg min { y i > 0 : i ∈ S ( c ) } . 3 Necessary conditions for stabilit y T o in tro duce the main ideas, consider first a simpler finite and deterministic problem. Let ( C, S, E ) b e a matching graph. Consider a batc h of customers x ∈ N C and a batc h of servers y ∈ N S of equal size: P c x c = P s y s . A p erfe ct matching of x and y is a tuple m ∈ N E suc h that: ∀ c ∈ C , x c = X s ∈ S ( c ) m cs , ∀ s ∈ S, y s = X c ∈ C ( s ) m cs . By Hall’s Theorem (ak a the “marriage Theorem”), there exists a perfect matc hing if and only if: P c ∈ U x c ≤ P s ∈ S ( U ) y s , ∀ U ⊂ C P s ∈ V y s ≤ P c ∈ C ( V ) x c , ∀ V ⊂ S (4) A p erfect matching, if there is one, can b e obtained by restating the mo del as a flow net work and b y solving the maxim um flo w problem for whic h efficien t algorithms exist [4, 3]. The bipartite matching mo del is m uch more complicated: first it is random, and second the matc hings hav e to b e p erformed on the fly , at each time step. How ever the tw o ingredien ts of the simpler mo del will play an instrumental role in the analysis: (i) the conditions NCond , to b e defined in (5), are related to (4); (ii) the restatemen t as a flow problem is used in most of the pro ofs. 7 Consider no w a bipartite matc hing mo del [( C, S, E ) , µ, Pol ]. W e identify the mo del with the Mark ov chain on the state space E describing the evolution of the buffer con tent. Let P be the transition matrix of the Mark ov chain. A probability measure π on E is stationary if πP = π . It is attr active if for any probabilit y measure ν on E , the sequence of Cesaro a verages of ν P n con verges weakly to π . Definition 3.1. The mo del is said to b e stable if the Markov chain has a unique and attr active stationary pr ob ability me asur e. It implies in particular that the graph of the Marko v chain has a unique terminal strongly connected comp onent with all states leading to it. Let µ C b e a probabilit y measure on C and µ S a probabilit y measure on S . Define the following conditions on ( µ C , µ S ): NCond :  µ C ( U ) < µ S ( S ( U )) , ∀ U ( C µ S ( V ) < µ C ( C ( V )) , ∀ V ( S (5) The ab o ve conditions app ear in [2]. They hav e a natural interpretation. Let µ C and µ S b e the marginals of the arriv al probability µ . Customers from U need to b e matc hed with serv ers from S ( U ). The first line in NCond asks for strictly more servers in a verage from S ( U ) than customers from U . The second line has a dual interpretation. Using the Strong La w of Large Num b ers, we also see that the arriv als up to time n satisfy (4) for all v alues of n large enough if and only if NCond is satisfied. Lemma 3.2. The c onditions NCond ar e ne c essary stability c onditions: if the Markov chain is stable then the c onditions NCond ar e satisfie d by the mar ginals of µ . Pr o of. W e suppose that the conditions NCond are not satisfied. Assume first that there exists U ⊂ C such that µ C ( U ) > µ S ( S ( U )). Let A n , resp. B n , b e the total n umbers of customers of type U , resp. servers of type S ( U ), to arrive in the system up to time n . Let X n b e the num b er of customers of type U present in the system at time n . By definition, X n ≥ A n − B n . By the Strong Law of Large Numbers, w e ha v e, a.s., lim n A n n = µ C ( U ) , lim n B n n = µ S ( S ( U )) , lim n X n n ≥ µ C ( U ) − µ S ( S ( U )) > 0 . So the Mark ov c hain is transien t. Similarly , if there exists V ⊂ S such that µ S ( V ) > µ C ( C ( V )), the mo del is unstable. (This part of the argumen t appears in [2, Prop. 3.4].) Assume now that there exists U ⊂ C, U 6 = C, suc h that µ C ( U ) = µ S ( S ( U )) . (6) Observ e that S ( U ) 6 = S , otherwise w e would ha ve µ C ( U ) = µ S ( S ) = 1 whic h would con tradict U 6 = C . Set V = S − S ( U ). Eqn (6) is equiv alent to: µ S ( V ) = µ C ( C − U ). The bipartite matching graph ( C , S, E ) is represented in Figure 5. By assumption, U × V ∩ E = ∅ . S V S(U) C U C−U Figure 5: The bipartite graph ( C, S, E ). W e ha ve µ ( U × V ) = µ C ( U ) − µ ( U × S ( U )) µ (( C − U ) × S ( U )) = µ S ( S ( U )) − µ ( U × S ( U )) . 8 Using Eqn (6), w e get µ ( U × V ) = µ (( C − U ) × S ( U )) . (7) Let D n b e the num b er of departures of t yp e ( C − U ) × S ( U ) up to time n . Let A n , B n , and X n b e defined as abov e. Set Z n = A n − B n . F or an arriv al of t yp e U × V , the Z -pro cess mak es a +1 jump, for an arriv al of t yp e ( C − U ) × S ( U ), the Z -pro cess makes a -1 jump, otherwise the Z -process remains constan t. W e ha ve X n ≥ A n − ( B n − D n ) ≥ Z n . W e ha ve t wo cases: - If µ ( U × V ) > 0, then, according to (7), the Z -pro cess is null recurren t. - If µ ( U × V ) = 0, then for an y initial condition suc h that X 0 > 0, a.s. X n ≥ X 0 , ∀ n . Hence, in b oth cases, the mo del cannot b e stable. Remark 2. Consider a non-c onne cte d matching graph ( C, S, E ). Consider a probability µ and an admissible matching p olicy such that the bipartite matching mo del is stable. Let ( C 0 , S 0 , E 0 ) b e a connected subgraph of ( C , S, E ). F ollowing the exact same steps as in the proof of Lemma 3.2, we prov e that µ C ( C 0 ) = µ S ( S 0 ) , µ ( C 0 × ( S − S 0 )) = 0 , µ (( C − C 0 ) × S 0 ) = 0 (otherwise the Mark ov c hain is either transien t or null recurrent). Therefore, we can decom- p ose the mo del into connected comp onen ts and treat them separately . Hence the assumption of connectedness of ( C, S, E ) in Def. 2.1 was made without loss of generality . 3.1 Complexit y of v erifying NCond Let us fix ( C, S, E ) and the probabilit y measures ( µ C , µ S ) suc h that supp( µ C ) = C, supp( µ S ) = S . W e w ant an efficien t algorithm to decide if the conditions NCond are satisfied. The n umber of inequalities in NCond is exp onen tial in | C | + | S | . So chec king directly if all the inequalities are satisfied is a metho d whose time complexity is exp onen tial in | C | + | S | . T o go b ey ond, we need additional material. W e use the standard terminology of netw ork flo w theory , see for instance [4]. Consider the directed graph N =  C ∪ S ∪ { i, f } , E ∪ { ( i, c ) , c ∈ C } ∪ { ( s, f ) , s ∈ S }  . (8) Endo w the arcs of E with infinite capacity , an arc of t yp e ( i, c ) with capacity µ C ( c ), and an arc of type ( s, f ) with capacit y µ S ( s ). i S C f Figure 6: The graph N asso ciated with the NN mo del of Figure 2. Recall that a cut is a subset of the arcs whose remov al disconnects i and f . The c ap acity of a cut is the sum of the capacities of the arcs. Set A = E ∪ { ( i, c ) , c ∈ C } ∪ { ( s, f ) , s ∈ S } . Recall that T : A → R + is a flow if: (i) ∀ c, T ( i, c ) = P s ∈ S ( c ) T ( c, s ) , ∀ s, P c ∈ C ( s ) T ( c, s ) = T ( s, f ); (ii) ∀ ( x, y ) ∈ E , T ( x, y ) is less or equal to the capacity of ( x, y ). The value of T is P c T ( i, c ) = P s T ( s, f ). Let NCond ≤ b e the set of inequalities obtained from NCond by replacing the strict inequal- ities by large inequalities. 9 Lemma 3.3. Ther e exists a flow of value 1 in N iff ( µ C , µ S ) satisfies NCond ≤ . Ther e exists a flow T of value 1 such that T ( c, s ) > 0 for al l ( c, s ) ∈ E iff ( µ C , µ S ) satisfies NCond . The first part of Lemma 3.3 is prov ed in [2, Prop. 3.7]. W e rep eat the argumen t for complete- ness. Pr o of. The celebrated Max-flo w Min-cut Theorem [4] states that the maximal v alue of a flo w is equal to the minimal capacit y of a cut. Observ e that the set of arcs { ( i, c ) , c ∈ C } forms a cut of capacit y 1. Therefore the maximal flo w is ≤ 1 and it is 1 iff all cuts ha ve a capacit y ≥ 1. T o b e of finite capacit y , a cut must con tain only customer arcs { ( i, c ) , c ∈ C } and server arcs { ( s, f ) , s ∈ S } . Consider a subset A = { ( i, c ) , c ∈ C 1 } ∪ { ( s, f ) , s ∈ S 1 } . Set C 2 = C − C 1 and S 2 = S − S 1 . The set A is a cut iff C 2 × S 2 ∩ E = ∅ , equiv alently iff S ( C 2 ) ⊂ S 1 and C ( S 2 ) ⊂ C 1 . Also the capacity of A is µ C ( C 1 ) + µ S ( S 1 ). C1 C2 S1 S2 Figure 7: Illustration of the proof of Lemma 3.3. Assume that the cut { ( i, c ) , c ∈ C 1 } ∪ { ( s, f ) , s ∈ S 1 } is of capacity strictly less than 1. W e ha ve µ C ( C 1 ) + µ S ( S 1 ) < 1 ⇐ ⇒ µ C ( C 1 ) < µ S ( S 2 ) . But C ( S 2 ) ⊂ C 1 so, if NCond ≤ is satisfied, we m ust ha ve: µ S ( S 2 ) ≤ µ C ( C ( S 2 )) ≤ µ C ( C 1 ) . So we hav e pro ved that NCond ≤ is not satisfied. The other w ay round, if NCond ≤ is not satisfied, then there exist C 1 , S 2 , C ( S 2 ) = C 1 suc h that µ C ( C 1 ) < µ S ( S 2 ). Set C 2 = C − C 1 and S 1 = S − S 2 . By definition, C 2 × S 2 ∩ E = ∅ , therefore { ( i, c ) , c ∈ C 1 } ∪ { ( s, f ) , s ∈ S 1 } is a cut. Its capacit y is µ C ( C 1 ) + µ S ( S 1 ) < 1. By contrapposing the abov e, w e get that: h NCond ≤ satisfied i ⇐ ⇒ h all cuts hav e a capacit y ≥ 1 i ⇐ ⇒ h maximal flow is 1 i . W e now prov e the second part of the lemma. Assume that the conditions NCond are not satisfied. If the conditions NCond ≤ are not satisfied either, then by the first part of the proof there exists no flow of v alue 1. Assume now that the conditions NCond ≤ are satisfied. Then there exists U ⊂ C, U 6 = C , suc h that µ C ( U ) = µ S ( S ( U )). Let T b e any flow of v alue 1. Using the flow relation for U , w e get: X ( c,s ) ∈ U × S ( U ) T ( c, s ) = X ( i,c ) ∈{ i }× U T ( i, c ) = µ C ( U ) . Using µ C ( U ) = µ S ( S ( U )) and the flow relation for S ( U ), w e deduce that: h X ( c,s ) ∈ U × S ( U ) T ( c, s ) = µ S ( S ( U )) i = ⇒ h X ( c,s ) ∈ ( C − U ) × S ( U ) T ( c, s ) = 0 i . 10 No w it follo ws from the connectedness of ( C , S, E ) that ( C − U ) × S ( U ) ∩ E 6 = ∅ . W e conclude that the flow T is suc h that T ( c, s ) = 0 for some ( c, s ) ∈ E . Assume no w that the conditions NCond are satisfied. Fix η such that 0 < η < 1 / | E | . Consider the function T η : A → R + defined by T η ( x, y ) =      η for ( x, y ) = ( c, s ) ∈ E | S ( c ) | η for ( x, y ) = ( i, c ) | C ( s ) | η for ( x, y ) = ( s, f ) . By construction T η is a flow. Set e µ C ( c ) = µ C ( c ) − | S ( c ) | η 1 − | E | η , e µ S ( s ) = µ S ( s ) − | C ( s ) | η 1 − | E | η . (9) F or η small enough, observe that e µ C , resp. e µ S , is a probabilit y measure on C , resp. on S . Cho ose η small enough such that ( e µ C , e µ S ) satisfies NCond . This is p ossible since the conditions NCond are op en conditions. Consider the directed graph N , see (8), with new capacities on the customer and server arcs defined b y e µ C and e µ S . By applying the first part of the proof, there exists a flow e T : A → R + of v alue 1. Define T : A → R + , T = T η + (1 − | E | η ) e T . By construction T is a flow for the graph N with the original capacity constrain ts ( µ C for the customer arcs and µ S for the server arcs). The v alue of T is 1 and it satisfies T ( x, y ) > 0 for all ( x, y ) ∈ E . This completes the pro of. There exist algorithms to find the maximal flo w which are p olynomial in the size of the un- derlying graph, indep endent of the arc capacities. F or instance, the classical “augmenting path algorithm” of Edmonds & Karp [3] op erates in O (( | C | + | S | ) | E | 2 ) time, and there exist more sophisticated algorithms op erating in O (( | C | + | S | ) 3 ) time. T ake one of these polynomial algorithms, call it MaxFlow and consider it as a blackbox. W e build on this to design a p olynomial algorithm to chec k NCond . Let us detail the construction. Lemma 3.4. Define ( e µ C , e µ S ) as in (9). The p air ( µ C , µ S ) satisfies NCond iff the p air ( e µ C , e µ S ) satisfies NCond for η strictly p ositive and smal l enough. Pr o of. Assume that ( µ C , µ S ) satisfies NCond . Since we are dealing with open conditions, an y small enough p erturbation of ( µ C , µ S ) still satisfies NCond . Assume no w that ( µ C , µ S ) does not satisfy NCond . There exists U ⊂ C , U 6 = C, such that µ C ( U ) ≥ µ S ( S ( U )). By using (9), we get  1 − | E | η  e µ C ( U ) +  X c ∈ U | S ( c ) |  η ≥  1 − | E | η  e µ S ( S ( U )) +  X s ∈ S ( U ) | C ( s ) |  η  1 − | E | η  e µ C ( U ) +   E ∩ ( U × S ( U ))   η ≥  1 − | E | η  e µ S ( S ( U )) +   E ∩ ( C ( S ( U )) × S ( U ))   η . By definition, we hav e U ⊂ C ( S ( U )). W e conclude that e µ C ( U ) ≥ e µ S ( S ( U )). So the pair ( e µ C , e µ S ) do es not satisfy NCond . Using Lemmas 3.3 and 3.4, NCond is satisfied iff MaxFlow ( N , e µ C , e µ S ) returns 1 for η small enough. So the tric k is to run MaxFlow on the input ( N , e µ C , e µ S ) b y considering η as a formal parameter made “as small as needed”. The precise meaning is the following. If x 1 , x 2 , y 1 , y 2 ∈ R , then: ( x 1 + y 1 η ) + ( x 2 + y 2 η ) = ( x 1 + x 2 ) + ( y 1 + y 2 ) η . F urthermore,  x 1 + y 1 η = x 2 + y 2 η  ⇐ ⇒  x 1 = x 2 , y 1 = y 2   x 1 + y 1 η < x 2 + y 2 η  ⇐ ⇒ h ( x 1 < x 2 ) or ( x 1 = x 2 , y 1 < y 2 ) i (10) 11 So η is small enough not to rev erse an y strict inequality . When running MaxFlow on ( N , e µ C , e µ S ), the algorithm deals with v alues of the type ( x + y η ), and adds and compare them according to the ab o v e rules. Now observe that the algorithm stops in finite time, so it will ha ve p erformed only a finite n umber of op erations. Therefore, it would b e p ossible, a p osteriori, to assign to η a v alue whic h w ould be small enough to enforce (10). Algorithm 2: Checking the necessary stability conditions Data : ( C , S, E ), ( µ C , µ S ) such that supp( µ C ) = C , supp( µ S ) = S . Result : “Y es” if NCond , “No” if ¬ ( NCond ) b egin Compute N , e µ C , e µ S ; if MaxFlow ( N , e µ C , e µ S ) = 1 then Resul t ← Y es; else Resul t ← No; return Resul t ; The termination is ob vious and the correctness follows from Lemmas 3.3 and 3.4. Prop osition 3.5. Given a bip artite mo del [( C, S, E ) , µ ] , ther e exists an algorithm of time c om- plexity O (( | C | + | S | ) 3 ) to de cide if NCond is satisfie d. 4 Connectivit y prop erties of the Marko v chain Define the following prop ert y for the transition graph of the Mark ov chain: UTC : a unique (terminal) strictly connected component with all states leading to it. Prop ert y UTC is necessary for stability as defined in Def. 3.1. But prop ert y UTC is not granted in bipartite matc hing mo dels and counterexamples are given below (Examples 2 and 3). In fact, w e will see that w e are in an unusual situation: the necessary stability conditions NCond turn out to be sufficient conditions for the property UTC (Theorem 4.3)! Observe also that property UTC is w eaker than irreducibilit y , and we will give an example of a mo del satisfying NCond and UTC without being irreducible (Example 4). 4.1 Stable structures T o establish property UTC , w e make a detour by in tro ducing and studying a notion of independent in terest: stable structures. Definition 4.1. A bip artite matching structur e ( C, S, E , F ) is stable if ther e exists a pr ob ability me asur e µ satisfying (1) and whose mar ginals µ C and µ S satisfy NCond . The justification for this terminology will app ear in § 7: we prov e there that under the ML p olicy , any model satisfying NCond is stable. So a structure is stable iff there exists an asso ciated mo del which is stable. First of all, there exist stable structures. Example 1. Consider ( C , S, E , C × S ), where ( C , S, E ) is the NN bipartite graph of Figure 2. Let µ C : µ C (1) = µ C (2) = 2 / 5 , µ C (3) = 1 / 5 , µ S : µ S (1 0 ) = µ S (2 0 ) = 2 / 5 , µ S (3 0 ) = 1 / 5 . The product measure µ = µ C × µ S has marginals µ C and µ S and w e chec k that ( µ C , µ S ) satisfy NCond . Also it is easily prov ed that for an y admiss ible matc hing p olicy , the graph of the Marko v c hain is irreducible. 12 On the other hand, there exist unstable structures. W e illustrate this on tw o examples. Example 2. Consider the structure ( C , S, E , F ) where ( C, S, E ) is the NN graph of Figure 2, and where F =  (1 , 3 0 ) , (2 , 2 0 ) , (3 , 1 0 )  . Consider any µ with supp( µ ) = F . W e ha ve µ C (1) = µ S (3 0 ) = µ (1 , 3 0 ) which violates NCond for V = { 3 0 } . W e can also prov e that the prop ert y UTC is not satisfied. Consider a state of the type ( x, y ) with x = y = (0 , 0 , k ), for som e k ≥ 0. An y one of the three possible arriv als leav e the state unc hanged. In particular, there is an infinite num b er of terminal components. Example 3. Consider the bipartite matc hing structure defined in Figure 8. The graph ( C, S, E ) is represented on the left of the figure, while the graph ( C , S, F ) is represen ted on the righ t. 1 2 3 4 1’ 2’ 3’ 4’ 1 2 3 4 1’ 2’ 3’ 4’ C S Figure 8: The matching graph ( C, S, E ) on the left, and the arriv al graph ( C , S, F ) on the righ t. Consider any µ with supp( µ ) = F . W e ha ve µ S ( { 1 0 , 2 0 } ) = µ (3 , 1 0 ) + µ (4 , 2 0 ) ≤ µ C ( { 3 , 4 } ) , whic h contradicts NCond for U = { 3 , 4 } . W e can also prov e that the prop erty UTC is not satisfied. Consider a state ( x, y ) with x 3 + x 4 = k > 0. Reducing the num b er of customers of t yp es 3/4 would require an arriv al of t yp e (1 , 1 0 ) or (1 , 2 0 ) or (2 , 1 0 ) or (2 , 2 0 ). But none of these pairs belong to F . Therefore it is imp ossible to reac h a state ( x 0 , y 0 ) with x 0 3 + x 0 4 < x 3 + x 4 . On the other hand an arriv al of type (3 , 3 0 ) or (3 , 4 0 ) or (4 , 3 0 ) or (4 , 4 0 ) strictly increases the num b er of customers of types 3/4. Hence all the states are transient, and there is no terminal strongly connected comp onent. Stabilit y of a structure is a decidable prop ert y . There exists a probabilit y measure µ with the requested properties iff the following system of linear inequalities in the indeterminates µ ( c, s ) , c ∈ C, s ∈ S, ha ve a solution:                    P ( c,s ) ∈ C × S µ ( c, s ) = 1 , µ ( c, s ) > 0 , ∀ ( c, s ) ∈ F , µ ( c, s ) = 0 , ∀ ( c, s ) ∈ C × S − F, µ C ( c ) = P s ∈ S µ ( c, s ) , ∀ c ∈ C , µ S ( s ) = P c ∈ C µ ( c, s ) , ∀ s ∈ S, NCond . (11) Ho wev er, the num b er of inequalities is exp onential in | C | + | S | . W e are going to propose a criterion whic h is muc h simpler, b oth conceptually and algorithmically . Consider a bipartite matc hing structure ( C , S, E , F ). Define e F = { ( s, c ) | ( c, s ) ∈ F } . Associate with the structure the directed graph ( C ∪ S, E ∪ e F ), in other w ords the no des are C ∪ S and the arcs are c − → s, if ( c, s ) ∈ E , s − → c, if ( c, s ) ∈ F . W e ha ve represen ted in Figure 9 the directed graph asso ciated with the structure of Example 3. 13 Figure 9: The directed graph associated with the structure of Figure 8. The graph of Figure 9 is not strongly connected: the four no des on the righ t form a strongly connected comp onent. Similarly , the directed graph asso ciated with the structure of Example 2 is not strongly connected. On the other hand, the directed graph asso ciated with the structure of Example 1 is strongly connected. This is not a coincidence. Theorem 4.2. L et ( C , S, E , F ) b e a bip artite matching structur e. The fol lowing two pr op erties ar e e quivalent: 1. ( C, S, E , F ) is a stable structur e; 2. ( C ∪ S, E ∪ e F ) is str ongly c onne cte d. In particular, one can decide if a structure is stable with an algorithm of time complexit y O ( | C || S | ) b y testing the strong connectivit y of ( C ∪ S, E ∪ e F ). Pr o of of The or em 4.2. Assume that ( C , S, E , F ) is a stable structure. Let µ b e a probability measure satisfying (1) and NCond . Suppose that there exist c ∈ C , s ∈ S , with no directed path from c to s in ( C ∪ S, E ∪ e F ). Let succ( c ) be the set of no des that can b e reac hed starting from c in ( C ∪ S, E ∪ e F ). Set C 1 = C ∩ succ( c ) , S 1 = S ∩ succ( c ) , C 2 = C − C 1 , S 2 = S − S 1 . By assumption, s ∈ S 2 . The follo wing t wo prop erties hold: µ ( C 2 , S 1 ) = 0 , ( C 1 × S 2 ) ∩ E = ∅ . Using µ ( C 2 , S 1 ) = 0, we get µ S ( S 1 ) = µ ( C 1 , S 1 ) ≤ µ C ( C 1 ) . But using [( C 1 × S 2 ) ∩ E = ∅ ] and NCond for U = C 1 , we get µ C ( C 1 ) < µ S ( S ( C 1 )) = µ S ( S 1 ) . F rom this con tradiction, w e deduce that for all c ∈ C, s ∈ S , there exists a directed path from c to s in ( C ∪ S, E ∪ e F ). Similarly , w e can prov e that for all s ∈ S, c ∈ C , there exists a directed path from s to c in ( C ∪ S, E ∪ e F ). Assume now that ( C ∪ S, E ∪ e F ) is strongly connected. Consider the matrices A ∈ R C × S + and B ∈ R S × C + defined by A cs = ( 1 / | S ( c ) | if ( c, s ) ∈ E 0 otherwise , B sc = ( 1 / # { d, ( s, d ) ∈ e F } if ( s, c ) ∈ e F 0 otherwise . Consider the matrix AB ∈ R C × C + . By construction, w e hav e ( AB ) cd > 0 if and only if there is a path of length 2 from c to d in the graph ( C ∪ S , E ∪ e F ). Since ( C ∪ S, E ∪ e F ) is strongly connected, we deduce that AB is irreducible. Clearly the sp ectral radius of AB is 1. Applying 14 the P erron-F rob enius Theorem [5], we obtain the existence of a line vector x ∈ R C + suc h that: ∀ c, x c > 0, P c x c = 1, and xAB = x . Set y = xA . Define the probability measure µ on C × S b y µ ( c, s ) = y s B sc . By construction, we hav e µ C = x, µ S = y . Also, b y construction, supp( µ ) = F . Define the function T : A → R + b y ∀ c ∈ C , T ( i, c ) = x c , ∀ s ∈ S, T ( s, f ) = y s , ∀ ( c, s ) ∈ E , T ( c, s ) = x c A cs . By construction, T is a flow of v alue 1 such that T ( c, s ) > 0 for all ( c, s ) ∈ E . Using Lemma 3.3, w e get that ( x, y ) = ( µ C , µ S ) satisfies NCond . 4.2 Bac k to prop ert y UTC W e no w ha ve all the ingredients needed to prov e the following result. Theorem 4.3. Consider a bip artite matching mo del [( C, S, E ) , µ, Pol ] . Assume that the structur e ( C, S, E , F ) is stable, e quivalently that ( C ∪ S, E ∪ e F ) is str ongly c onne cte d. Then the tr ansition gr aph of the Markov chain of the bip artite matching mo del satisfies the pr op erty UTC . Pr o of of The or em 4.3. W e are going to pro ve that the empty state can be reac hed starting from an y state. This is a sufficien t condition for prop ert y UTC to hold. The unique terminal strongly connected comp onent is the set of states that can b e reached from the empty state. W e carry out the proof in the commutativ e case, but it w orks unchanged in the non-commutativ e case (the only information needed is the n umber of customer/server of eac h class). Consider a non-empt y state ( X , Y ), with X = ( x c ) c ∈ C and Y = ( y s ) s ∈ S . It is sufficient to pro ve that w e can alw ays reach a state ( X 0 , Y 0 ) such that | X 0 | < | X | . If there exists ( c, s ) ∈ C } × S } suc h that µ ( c, s ) > 0, then the pro of is completed. Assume no w that µ ( C } × S } ) = 0. Cho ose ( c, s ) ∈ C } × S } . By assumption, there exists a path from s to c in ( C ∪ S, E ∪ e F ). Let us denote it by ( s = s 1 , c 1 , s 2 , c 2 , . . . , s k , c k = c ). Assume that c 1 , . . . , c k − 1 6∈ C } and s 2 , . . . , s k 6∈ S } . (If not, consider a subpath with this prop ert y .) Assume further that c 1 ∈ C ◦ . (If c 1 ∈ C • , then ( c 1 , s 2 ) ∈ E implies s 2 ∈ C } , so we can consider the subpath ( s = s 2 , c 2 , . . . , s k , c k = c ).) Since ( s i , c i ) ∈ e F and ( c i , s i +1 ) ∈ E b y construction and since c 1 ∈ C ◦ , we get that c 1 , . . . , c k − 1 ∈ C ◦ and s 2 , . . . , s k ∈ S ◦ . s k c k e F E e F e F e F E e F ( c 1 , s 1 ) s 1 c 1 Figure 10: The path ( s = s 1 , c 1 , s 2 , c 2 , . . . , s k , c k = c ) in ( C ∪ S, E ∪ e F ). By definition of the graph ( C ∪ S, E ∪ e F ), we ha ve µ ( c i , s i ) > 0 for all i . Cho ose the sequence of arriv als ( c 1 , s 1 ) , . . . , ( c k , s k ). Consider the effect of the arriv al of ( c 1 , s 1 ). Since s 1 ∈ S } , it will be matc hed with a customer of C • (and not with c 1 , ev en if ( c 1 , s 1 ) ∈ E , since an admissible matc hing policy is alw ays buffer-first , see § 2.2). Since c 1 6∈ C } , it will remain unmatched. Let ( X (1) , Y (1) ) b e the new state. W e ha ve | X (1) | = | X | . Also, in the new state, w e hav e c 1 ∈ C (1) • , whic h implies that s 2 ∈ S (1) } . So we can repeat the argumen t inductively . After the arriv als of ( c 1 , s 1 ) , . . . , ( c k − 1 , s k − 1 ), we are in a state ( X ( k − 1) , Y ( k − 1) ) satisfying: | X ( k − 1) | = | X | , s k ∈ S ( k − 1) } , c k ∈ C ( k − 1) } . Therefore, after the arriv al of ( c k , s k ), we end up in a state ( X ( k ) , Y ( k ) ) suc h that | X ( k ) | = | X | − 1. This completes the proof. 15 Example 4. Consider a bipartite matching model associated with the structure ( C , S, E , F ) where ( C, S, E ) is the NN graph of Figure 2, and where F =  (1 , 1 0 ) , (2 , 2 0 ) , (3 , 3 0 )  . The graph ( C ∪ S, E ∪ e F ) is strongly connected. According to Theorem 4.2, the graph satisfies prop- ert y UTC . But it is not irreducible. Indeed, it is impossible to reach the state ((0 , 1 , 0); (0 , 0 , 1)) starting from the empty state. More generally , none of the states of the facet ( { 2 } , { 3 0 } ) b elong to the terminal strongly connected component. Belo w, we study the stability of bipartite matc hing mo dels. Therefore, w e alwa ys assume that the necessary conditions NCond are satisfied. So we get the prop ert y UTC for the Marko v chain as a consequence of Theorem 4.3. 5 Mo dels that are stable for all admissible p olicies Definition 5.1. Consider a bip artite gr aph ( C , S, E ) and an admissible matching p olicy Pol . The stabilit y region is the set of values of µ for which the bip artite matching mo del [( C, S, E ) , µ, Pol ] is stable. The stabilit y region is included in the p olyhedron defined b y NCond . The stability region is maximal if it is equal to this polyhedron. Denote by F the set of facets. Define the following conditions on µ : SCond : µ C ( C } ( F )) + µ S ( S } ( F )) > 1 − µ ( E ∩ C ◦ ( F ) × S ◦ ( F )) , ∀F ∈ F − {∅} (12) Let F b e a saturated facet, see Definition 2.4. Assume for instance that C ◦ ( F ) = ∅ . Then E ∩ C ◦ ( F ) × S ◦ ( F ) = ∅ and C } ( F ) = C − C • ( F ). So (12) implies: µ S ( S } ( F )) > µ C ( C • ( F )) . Since S } ( F ) = S ( C • ( F )), we recognize exactly (5) for U = C • ( F ). Conv ersely , consider U ( C and the asso ciated condition in NCond : µ C ( U ) < µ S ( S ( U )). Cho ose a state with a strictly p ositiv e num b er of customers/servers for the classes U and S − S ( U ). Let F be the corresp onding facet. The facet F is saturated: S } ( F ) = S ( U ) , S • ( F ) = S − S ( U ) , S ◦ ( F ) = ∅ . Let us apply (12) to the facet F , w e get: µ C ( C } ( F )) + µ S ( S ( U )) > 1 µ S ( S ( U )) > µ C ( C − C } ( F )) ≥ µ C ( U ) . T o summarize, the subset of the inequalities (12) obtained by considering only the saturated facets gives precisely the inequalities NCond . W e no w sho w that the conditions SCond are sufficient stability conditions. Prop osition 5.2. A bip artite mo del with pr ob ability µ satisfying SCond is stable under any admissible matching p olicy. Pr o of. Consider the linear Lyapuno v function: L ( u, v ) = | u | , ( u, v ) ∈ E , the n umber of unmatched customers (serv ers). Let ( U n , V n ) n b e the Mark o v chain of the buffer- con tent. Let F 6 = ∅ be an arbitrary and fixed facet. Then for any ( u, v ) ∈ F we ha v e (see T able 1): E[ L ( U n +1 , V n +1 ) | ( U n , V n ) = ( u, v )] − L ( u, v ) = − µ ( C } ( F ) , S } ( F )) + µ ( C ◦ ( F ) , S • ( F )) + µ ( C • ( F ) , S • ( F )) + µ ( C • ( F ) , S ◦ ( F )) + µ ( C ◦ ( F ) × S ◦ ( F ) ∩ E c ) = 1 − µ C ( C } ( F )) − µ S ( S } ( F )) − µ ( C ◦ ( F ) × S ◦ ( F ) ∩ E ) . 16 The inequality (12) implies directly that: E[ L ( U n +1 , V n +1 ) | ( U n , V n ) = ( u, v )] − L ( u, v ) <  < 0 . (13) By application of the Lyapuno v-F oster Theorem, see for instance [1, § 5.1], we conclude that the mo del is stable. C } C ◦ C • S } − 1 0 0 S ◦ 0 0 or 1 1 S • 0 1 1 T able 1: V ariation of the linear Lyapuno v function. Corollary 5.3. Consider a bip artite gr aph in which any non-zer o fac et is satur ate d. F or any admissible matching p olicy, the stability r e gion is maximal. The bipartite graph ( C = { 1 , 2 } , S = { 1 0 , 2 0 } , C × S − { (2 , 2 0 ) } ) is such that any non-zero facet is saturated. Therefore, its stability region is maximal for an y admissible p olicy . The same is true for the “almost complete graphs” ( C = { 1 , . . . , k } , S = { 1 0 , . . . , k 0 } , C × S − { ( i, i 0 ) , ∀ i } ). Example 5. Consider the NN graph from Figure 2. The graph has only one non-zero facet that is non-saturated, facet ( { 3 } , { 3 0 } ). F or an y admissible p olicy , the stability region is at least the p olyhedron SCond , Proposition 5.2, which is defined by: NCond , µ C (1) + µ S (1 0 ) > 1 − µ (2 , 2 0 ) . (14) Assume now µ = µ C × µ S and µ C = µ S . Set x = µ C (1) = µ S (1 0 ) and y = µ C (2) = µ S (2 0 ). Then: NCond :  x < 0 . 5 2 x + y > 1 SCond :  NCond 2 x + y 2 > 1 In Figure 11, the light (yello w) region corresp onds to SCond , and the union of the light and Figure 11: NCond and SCond for the NN-graph with µ = µ C × µ S and µ C = µ S . dark (red) regions corresponds to NCond . Unfortunately , for some bipartite graphs, the p olyhedron SCond is empty . This is illustrated b y the following example. Example 6. Consider the NNN graph of Figure 12. The condition SCond for facet ( { 1 } , { 4 0 } ) giv es: µ C ( { 3 , 4 } ) + µ S ( { 1 0 , 2 0 } ) > 1 − µ (2 , 3 0 ) (15) and for facet ( { 4 } , { 1 0 } ): µ C (1) + µ S (4 0 ) > 1 − µ (2 , 2 0 ) − µ (2 , 3 0 ) − µ (3 , 3 0 ) . (16) 17 The inequalit y (15) is equiv alen t to: µ C (1) + µ S (4 0 ) < 1 − µ C (2) − µ ( { 1 , 3 , 4 } , 3 0 ) . T ogether with (16) this gives: µ C (1) + µ S (4 0 ) < 1 − µ C (2) − µ ( { 1 , 3 , 4 } , 3 0 ) < 1 − µ (2 , 2 0 ) − µ (2 , 3 0 ) − µ (3 , 3 0 ) < µ C (1) + µ S (4 0 ) , whic h is imp ossible. 1 3 2 2’ 4 4’ 1’ 3’ 1 3 2 2’ 4 4’ 1’ 3’ Figure 12: NNN graph: facets ( { 1 } , { 4 0 } ) and ( { 4 } , { 1 0 } ). {3},{2’} 0 {4},{1’,2’} {4},{2’} {4},{1’} {4},{3’} {3},{1’} {2},{1’} {2},{4’} {1},{3’} {1},{4’} {4},{1’,2’,3’} {3,4},{1’,2’} {3,4},{1’} {3,4},{2’} {1},{3’,4’} {1,2},{4’} {1,4},{3’} {2},{1’,4’} {2,3},{1’} {2,4},{1’} {3},{1’,2’} {2,3,4},{1’} {4},{2’,3’} {4},{1’,3’} Figure 13: F acets for the NNN graph. Saturated facets are encircled (13 among 25 facets). 6 Priorities and MS are not alw a ys stable Consider the NN bipartite graph of Figure 2 and Example 5. F or this mo del, Proposition 5.2 do es not allo w to decide if the stability region is maximal (see Figure 11). In Figure 14, w e give sim ulation results for the a verage buffer size up to time n = 1000000 for the NN-graph with µ = µ C × µ S , µ C = µ S , and MS p olicy . W e can see that the av erage buffer size is growing rapidly Figure 14: Av erage buffer size for the NN-graph with µ = µ C × µ S , µ C = µ S , and MS policy . near the 2 x + y = 1 line. This do es not necessarily imply unstability , as ev en for stable mo dels w e could hav e the mean stationary buffer size that is growing unboundedly as we approach the b oundary of the stability region. In fact, w e sho w below that for the PR and MS matc hing policies, the stability region is not maximal. 18 Prop osition 6.1. Consider the NN mo del with either the MS p olicy or the PR p olicy given by: A =   0 2 1 2 1 0 1 0 0   and B =   0 2 1 2 1 0 1 0 0   . F or b oth p olicies, the stability r e gion is not maximal. Pr o of. W e carry out the proof for the PR p olicy . The idea of the proof is to pla y with t wo different Mark ov chains: the one describing the evolution of the buffer conten t, and an auxiliary one which mimic ks the evolution of the customers/servers of t yp e 2/2’ on some of the facets. Consider an auxiliary Mark o v c hain on Z with transition probabilities: x → x − 1 x → x x → x + 1 x < 0 a − 1 a 0 a 1 x = 0 b − 1 b 0 b 1 x > 0 c − 1 c 0 c 1 Assume that a − 1 , a 1 , b − 1 , b 1 , c − 1 , c 1 , are all different from 0. The chain is p ositiv e recurrent iff:  a − 1 < a 1 , c 1 < c − 1  . The stationary distribution is then equal to: π (0) =  1 + b − 1 a 1 − a − 1 + b 1 c − 1 − c 1  − 1 , π ( x ) = π (0) b 1 c 1  c 1 c − 1  x if x > 0 , π ( x ) = π (0) b − 1 a − 1  a − 1 a 1  | x | if x < 0 . Consider a NN-mo del with a probability µ such that supp( µ ) = C × S . Let ( X, Y ) = ( X ( n ) , Y ( n )) n b e the Mark ov chain of the buffer conten t, where X ( n ) = ( X 1 ( n ) , X 2 ( n ) , X 3 ( n )) and Y ( n ) = ( Y 1 ( n ) , Y 2 ( n ) , Y 3 ( n )). Assume wlog that ( X , Y ) is given under the form of a Stochastic Recursiv e Sequence, that is: ( X ( n + 1) , Y ( n + 1)) = Φ( X ( n ) , Y ( n ) , θ n ) , where ( θ n ) n is an i.i.d. sequence of r.v.’s distributed according to µ , and Φ is a deterministic function. Consider now the process ( X 2 ( n ) − Y 2 ( n )) n . This is not a Mark ov chain. Ho wev er, if X 2 ( n ) + X 3 ( n ) > 0, then “it b ecomes one”. More precisely , if X 2 ( n ) + X 3 ( n ) > 0, then X 2 ( n + 1) − Y 2 ( n + 1) = Φ 2 ( X 2 ( n ) − Y 2 ( n ) , θ n ) , (17) where Φ 2 is a deterministic function. This can b e chec ked b y direct insp ection. Moreo ver, the transition kernel on Z defined b y the recursion (17) is of the type of the ab o ve auxiliary chain with parameters: a 1 = µ (1 , 1 0 ) + µ (1 , 3 0 ) + µ (2 , 1 0 ) + µ (2 , 3 0 ) , a 0 = µ (1 , 2 0 ) + µ (2 , 2 0 ) + µ (3 , 1 0 ) + µ (3 , 3 0 ) , a − 1 = µ (3 , 2 0 ) , b 1 = µ (2 , 1 0 ) + µ (2 , 3 0 ) , b 0 = µ (1 , 1 0 ) + µ (1 , 3 0 ) + µ (2 , 2 0 ) + µ (3 , 1 0 ) + µ (3 , 3 0 ) , b − 1 = µ (1 , 2 0 ) + µ (3 , 2 0 ) , c 1 = µ (2 , 3 0 ) , c 0 = µ (1 , 3 0 ) + µ (2 , 1 0 ) + µ (2 , 2 0 ) + µ (3 , 3 0 ) , c − 1 = µ (1 , 1 0 ) + µ (1 , 2 0 ) + µ (3 , 1 0 ) + µ (3 , 2 0 ) . 19 Let us justify for instance the v alues of c − 1 , c 0 , c 1 . W e are in the case X 2 ( n ) + X 3 ( n ) > 0 and X 2 ( n ) − Y 2 ( n ) > 0 which implies: X 1 ( n ) = 0 , X 2 ( n ) > 0 , Y 1 ( n ) = Y 2 ( n ) = 0 , Y 3 ( n ) > 0 , In T able 2 b elo w, we show the effect of the differen t p ossible types of arriv als, restricting to the ones which may affect X 2 , i.e. when the customer class is 2 or the serv er class is 1 0 or 2 0 . T o simplify , w e ha v e assumed in T able 2 that X 3 > 0. F or X 3 = 0, the “P ossible matc hings” column w ould be affected, but not the “Selected matc hings” and “∆ X 2 ” columns. Arriv al P ossible matc hings Selected matchings ∆ X 2 (1 , 1 0 ) (1 , 3 0 ), (2 , 1 0 ), (3 , 1 0 ) (1 , 3 0 ), (2 , 1 0 ) -1 (1 , 2 0 ) (1 , 3 0 ), (1 , 2 0 ), (2 , 2 0 ) (1 , 3 0 ), (2 , 2 0 ) -1 (2 , 1 0 ) (2 , 1 0 ), (3 , 1 0 ) (2 , 1 0 ) 0 (2 , 2 0 ) (2 , 2 0 ) (2 , 2 0 ) 0 (2 , 3 0 ) ∅ ∅ +1 (3 , 2 0 ) (2 , 2 0 ) (2 , 2 0 ) -1 (3 , 1 0 ) (2 , 1 0 ), (3 , 1 0 ) (2 , 1 0 ) -1 T able 2: Effect of arriv als. Let us comment on a couple of cases. If the arriv al is of type (1 , 1 0 ), then the selected matching is (2 , 1 0 ) rather than (3 , 1 0 ) due to the PR p olicy ( B 2 , 1 0 > B 3 , 1 0 ). If the arriv al is of type (1 , 2 0 ), the selected matc hing is (2 , 2 0 ) rather than (1 , 2 0 ) according to the buffer-first property of admissible p olicies, see § 2.2. The other cases are argued similarly . Let us introduce a new Mark ov chain ( W n ) n on Z defined b y: W n +1 = Φ 2 ( W n , θ n ) . (The pro cess ( W n ) n is different from the pro cess ( X 2 ( n ) − Y 2 ( n )) n . The former is alw ays defined according to the recursion (17) while the latter is defined according to (17) only for the n ’s suc h that X 2 ( n ) + X 3 ( n ) > 0. The former is Mark ovian while the latter is not.) Condition c 1 < c − 1 b ecomes µ C (2) < µ S (1 0 ) + µ S (2 0 ) and a − 1 < a 1 b ecomes µ S (2 0 ) < µ C (1) + µ C (2). Both conditions follo w from NCond . So the auxiliary c hain ( W n ) n is ergo dic and its stationary distribution π satisfies: π (0) =  1 + b − 1 a 1 − a − 1 + b 1 c − 1 − c 1  − 1 , π ( Z ∗ + ) = π (0) b 1 c − 1 − c 1 , π ( Z ∗ − ) = π (0) b − 1 a 1 − a − 1 , where Z ∗ + = { 1 , 2 , . . . } is the set of strictly p ositiv e integers and Z ∗ − = {− 1 , − 2 , . . . } is the set of strictly negative integers. F rom no w on, we fix an initial condition W 0 satisfying W 0 ∼ π , W 0 ⊥ ⊥ ( θ n ) n . Let us switch bac k to the Marko v c hain ( X, Y ). Set L ( n ) = X 2 ( n ) + X 3 ( n ) , ∆ L ( n ) = L ( n + 1) − L ( n ) . If L ( n ) > 0, then we chec k b y direct insp ection that: ∆ L ( n ) = Ψ( X 2 ( n ) − Y 2 ( n ) , θ n ) , where Ψ is a deterministic function. W e ha ve in particular α def = E[∆ L ( n ) | L ( n ) > 0 , Y 2 ( n ) > 0] = µ (3 , 2 0 ) + µ (3 , 3 0 ) − µ (1 , 1 0 ) − µ (2 , 1 0 ) , β def = E[∆ L ( n ) | L ( n ) > 0 , X 2 ( n ) = Y 2 ( n ) = 0] = µ (2 , 3 0 ) + µ (3 , 2 0 ) + µ (3 , 3 0 ) − µ (1 , 1 0 ) , γ def = E[∆ L ( n ) | L ( n ) > 0 , X 2 ( n ) > 0] = µ (2 , 3 0 ) + µ (3 , 3 0 ) − µ (1 , 1 0 ) − µ (1 , 2 0 ) . 20 Let us turn again to the auxiliary c hain ( W n ) n . By p erforming the computation, we get E[Ψ( W n , θ n )] = π ( Z ∗ − ) α + π (0) β + π ( Z ∗ + ) γ . The Ergo dic Theorem for Mark ov Chains, see for instance [1, § 3.4], gives: lim n 1 n n − 1 X i =0 Ψ( W n , θ n ) = π ( Z ∗ − ) α + π (0) β + π ( Z ∗ + ) γ a.s. Assume that π ( Z ∗ − ) α + π (0) β + π ( Z ∗ + ) γ > 0. Then we hav e: lim n n − 1 X i =0 Ψ( W n , θ n ) = + ∞ a.s. (18) Therefore, for each ε > 0, there exists K ε ≥ 0 such that P  min n ≥ 1 n − 1 X i =0 Ψ( W n , θ n ) > − K ε  ≥ 1 − ε . (19) Let us switc h bac k to the Marko v chain ( X ( n ) , Y ( n )) n . Cho ose the initial condition ( X (0) , Y (0)) suc h that X 2 (0) − Y 2 (0) = W 0 min( X 3 (0) , Y 3 (0)) = K ε , (20) where K ε is defined in (19). By construction, on the even t A = { min n ≥ 1 P n − 1 i =0 Ψ( W n , θ n ) > − K ε } , w e ha ve ∀ n, L ( n ) > 0 , X 2 ( n ) − Y 2 ( n ) = W n . So, on the ev en t A , we hav e L ( n ) = K ε + n − 1 X i =0 ∆ L ( i ) = K ε + n − 1 X i =0 Ψ( W i , θ i ) − → + ∞ . (21) W e conclude that the Marko v c hain ( X, Y ) of the NN-mo del is transient. W e no w sho w that the stabilit y region is not maximal, b y giving an example suc h that π ( Z ∗ − ) α + π (0) β + π ( Z ∗ + ) γ > 0. Consider µ C = (1 / 3 , 2 / 5 , 4 / 15), µ S = µ C , and µ = µ C × µ S . Th us conditions NCond are satisfied. How ev er, w e ha ve: a 1 = 11 25 , a 0 = 34 75 , a − 1 = 8 75 , b 1 = 6 25 , b 0 = 13 25 , b − 1 = 6 25 , c 1 = 8 75 , c 0 = 34 75 , c − 1 = 11 25 , and π (0) = 25 61 , π ( Z ∗ + ) = 18 61 , π ( Z ∗ − ) = 18 61 . This gives α = − 1 / 15 , β = 13 / 75 , γ = − 1 / 15, and, π ( Z ∗ − ) α + π (0) β + π ( Z ∗ + ) γ = 29 915 > 0 . This completes the proof. Consider no w the MS p olicy . Set L ( n ) = min( X 3 ( n ) − X 2 ( n ) , Y 3 ( n ) − Y 2 ( n )). The initial distribution can b e taken such that X 2 (0) − Y 2 (0) ∼ π , ( X 3 (0) − X 2 (0) = K ε if X 2 (0) > 0 Y 3 (0) − Y 2 (0) = K ε otherwise . Mo dulo these modifications, the pro of carries ov er unchanged. 21 7 ML is alw a ys stable In this section, w e sho w that the ML p olicy has a maximal stabilit y region. The idea of the pro of is as follows. Consider the quadratic Ly apunov function: L ( x, y ) = X c ∈ C x 2 c + X s ∈ S y 2 s , ( x, y ) ∈ E . (22) Observ e that the ML p olicy minimizes the v alue of this Ly apuno v function at eac h step. W e in tro duce an alternate p olicy that dep ends on the arriv al distribution µ . F or this p olicy , we manage to prov e that the quadratic Ly apunov function has a negative drift outside a finite region. Theorem 7.1. F or any bip artite gr aph, the ML p olicy has a maximal stability r e gion. Pr o of. W e introduce an alternate matching p olicy . This p olicy is admissible, corresp onds to a comm utative state space, but do es not b elong to the policies listed in § 2.2. It is a random p olicy and its sp ecificit y is to be facet dep endent. Let us describe the alternate p olicy on a non-empty facet F . Set C • = C • ( F ) , S • = S • ( F ) , C } = C } ( F ) , etc. T o describ e the matching policy , the only thing we ha ve to describ e is the wa y to matc h an arriving customer of class c ∈ C } , resp. server of class s ∈ S } . Let us concen trate first on a server of class s ∈ S } . F rom NCond : µ C ( C • ) < µ S ( S } ) , µ S ( S • ) < µ C ( C } ) . W e build a directed graph as in (8) but restricted to the nodes in C • and S } . F ormally , N F =  C • ∪ S } ∪ { i, f } , { E ∩ C • × S } } ∪ { ( i, c ) , c ∈ C • } ∪ { ( s, f ) , s ∈ S } }  . (23) Endo w the arcs of E ∩ C • × S } with infinite capacit y , an arc of type ( i, c ) with capacity µ C ( c ), and an arc of type ( s, f ) with capacity µ S ( s ). As in Lemma 3.3, NCond implies that the minimal cut of N F has capacity µ C ( C • ). Any maximal flow T is suc h that: ∀ c ∈ C • , T ( i, c ) = µ C ( c ), ∀ s ∈ S } , T ( s, f ) ≤ µ S ( s ). Let us pro ve that there exists a maximal flo w T suc h that: ∀ c ∈ C • , T ( i, c ) = µ C ( c ) ∀ s ∈ S } , T ( s, f ) < µ S ( s ) . Define e µ S on S } b y e µ S ( s ) = µ S ( s ) − η . Here η > 0 is chosen to be small enough so that: ∀ U ⊂ C • , µ C ( U ) < e µ S ( S ( U )) , ∀ V ⊂ S } , e µ S ( V ) < µ C ( C ( V )). This is p ossible since NCond are op en conditions. Consider the same netw ork as ab o ve but with the capacities e µ S ( s ) on the arcs ( s, f ). The minimal cut still has capacit y µ C ( C • ). A maximal flow T is such that: ∀ c ∈ C • , T ( i, c ) = µ C ( c ), ∀ s ∈ S } , T ( s, f ) ≤ e µ S ( s ) < µ S ( s ). Clearly , T is also a flow for the original net work. The serv er s ∈ S } is matc hed to c ∈ C • ∩ C ( s ) randomly , indep enden tly of the past, with probabilit y: P F sc = 1 µ S ( s ) h T ( c, s ) + µ S ( s ) − T ( s, f ) | C • ∩ C ( s ) | i . Let us chec k that this defines indeed a probabilit y: X c ∈ C • ∩ C ( s ) P F sc = 1 µ S ( s ) h µ S ( s ) − T ( s, f ) + X c ∈ C • ∩ C ( s ) T ( c, s ) i = 1 µ S ( s ) h µ S ( s ) − T ( s, f ) + T ( s, f ) i = 1 . 22 F or c ∈ C • , s ∈ S ( c ), set ε sc = ( µ S ( s ) − T ( s, f )) / | C • ∩ C ( s ) | . F or c ∈ C • , set ε c = P s ∈ S ( c ) ε sc . W e ha ve ε c > 0. Observe that: ∀ c ∈ C • , X s ∈ S ( c ) µ S ( s ) P F sc = µ C ( c ) + ε c . (24) Symmetrically , we define the directed graph of t yp e (23) but on the nodes C } and S • . W e build a maximal flo w on this new graph as abov e, and based on this flow, we define the probabilit y P F cs that a customer c ∈ C } is matched to a serv er s ∈ S • ∩ S ( c ). F or s ∈ S • , we define ε cs , c ∈ C ( s ), and ε s accordingly . W e ha v e ε s > 0. Let ( X ( n ) , Y ( n )) n b e the Mark ov c hain of the buffer-conten t of the mo del. Assume that ( X ( n ) , Y ( n )) = ( x, y ) ∈ F and let c ∈ C • . W e ha ve: (i) X ( n + 1) c = X ( n ) c − 1 iff: - the arriving customer is not of class c ; - the arriving server is of class s ∈ S ( c ); - the arriving server is matc hed with c (probabilit y P F sc ). This case happ ens with probabilit y α c = P s ∈ S ( c ) µ ( C − c, s ) P F sc . (ii) X ( n + 1) c = X ( n ) c + 1 iff: - the arriving customer is of class c ; - the arriving server is not matched with c . This may o ccur in tw o p ossible wa ys: either the arriving serv er is of class s 6∈ S ( c ), or the arriving serv er is of class s ∈ S ( c ) but is not matched with c (probabilit y 1 − P F sc ). This case happ ens with probabilit y β c = µ ( c, S − S ( c )) + P s ∈ S ( c ) µ ( c, s )(1 − P F sc ). (iii) Otherwise, X ( n + 1) c = X ( n ) c . Using (24), we get: X s ∈ S ( c ) µ ( C − c, s ) P F sc = X s ∈ S ( c ) µ S ( s ) P F sc − X s ∈ S ( c ) µ ( c, s ) P F sc = µ C ( c ) + ε c − X s ∈ S ( c ) µ ( c, s ) P F sc = µ ( c, S − S ( c )) + µ ( c, S ( c )) + ε c − X s ∈ S ( c ) µ ( c, s ) P F sc = µ ( c, S − S ( c )) + X s ∈ S ( c ) µ ( c, s )(1 − P F sc ) + ε c . Th us, α c = β c + ε c . Observe that β c < µ C ( c ). W e get, for ( x, y ) ∈ F and c ∈ C • : E  X ( n + 1) 2 c − X ( n ) 2 c | ( X ( n ) , Y ( n )) = ( x, y )  = β c (2 x c + 1) − α c (2 x c − 1) = 2 β c − ε c (2 x c − 1) < 2 µ C ( c ) − ε c x c . Let L be the quadratic Lyapuno v function (22). Define ∆ L ( n ) = L ( X n +1 , Y n +1 ) − L ( X n , Y n ). Set ε = min v ∈ C • ∪ S • ε v > 0. Then (the first term in the sum tak es care of the vertices in C − C • and S − S • ): E[∆ L ( n ) | ( X n , Y n ) = ( x, y )] < 2 + X c ∈ C • (2 µ C ( c ) − ε c x c ) + X s ∈ S • (2 µ S ( s ) − ε s y s ) < 2 + 2 µ C ( C • ) + 2 µ S ( S • ) − ε X c ∈ C • x c + X s ∈ S • y s ! < 6 − 2 ε X c ∈ C x c . 23 Fix δ > 0. If P c ∈ C x c > (6 + δ ) / 2 ε , then E[∆ L ( n )] < − δ . There are finitely many facets, so there is a finite set A ⊂ E suc h that ∀ ( x, y ) 6∈ A, E[∆ L ( n )] < − δ . (25) By the Lyapuno v-F oster’s Theorem, see for instance [1, § 5.1], the alternate matc hing p olicy is stable. Since the ML matching p olicy minimizes the v alue of the quadratic Ly apunov function, w e ha ve a fortiori that (25) holds for it. Therefore, the ML policy is also stable. Conclusion. Many op en questions remain. First, we do not know if the stability region is alwa ys maximal for the FIFO and Random p olicies. Numerical experiments seem to indicate that it is indeed the case. Second, for the MS and priority p olicies, we know that the stabilit y region is not alw ays maximal, but w e do not know how to compute it. Last, we would like to obtain sufficien t conditions for stabilit y , v alid for all admissible p olicies, and whic h are b etter than the ones of § 5. The program used to carry out the numerical experiments is av ailable on request from Ana Bu ˇ si ´ c. References [1] P . Br´ emaud. Markov chains: Gibbs fields, Monte Carlo simulation, and queues , v olume 31 of T exts in Applie d Mathematics . Springer-V erlag, New Y ork, 1999. [2] R. Calden tey , E.H. Kaplan, and G. W eiss. F CFS infinite bipartite matching of serv ers and customers. A dv. Appl. Pr ob ab , 41(3):695–730, 2009. [3] J. Edmonds and R. Karp. 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