On Existence of Limiting Distribution for Time-Nonhomogeneous Countable Markov Process

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1 Queueing Systems 46, , Kluwer Academic Publishers Manufactured in The Netherlands On Existence of Limiting Distribution for Time-Nonhomogeneous Countable Markov Process V ABRAMOV vyachesl@zahavnetil Department of Mathematics, The Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, and College of Judea and Samaria, Ariel, Israel R LIPTSER liptser@engtauacil Department of Electrical Engineering-Systems, Tel Aviv University, Tel Aviv, Israel Received 15 October 22; Revised 1 March 23 Abstract In this paper, sufficient conditions are given for the existence of iting distribution of a nonhomogeneous countable Markov chain with time-dependent transition intensity matrix The method of proof exploits the fact that if the distribution of random process Q = (Q t t is absolutely continuous with respect to the distribution of ergodic random process Q = (Q t t,then law Q t π, where π is the invariant measure of Q We apply this result for asymptotic analysis, as t,ofa nonhomogeneous countable Markov chain which shares iting distribution with an ergodic birth-anddeath process Keywords: countable Markov process, existence of the iting distribution, birth-and-death process 1 Introduction There is a large number of papers in the queueing literature devoted to analysis of state dependent and time dependent queueing systems as M t /M t /1andM t /M t /c and associated with their Markovian queueing networks (eg, see [23 27] An analysis of such queueing system is motivated by a wide spectra of practical problems well-known in the literature For instance, a simple example corresponding to the police dispatching problem is given in [26, p 6] (see also [15] for further discussion The M t /M t / queue, used for a model of emergency ambulances and intensive care units, is considered in [6] Other applications are known for client/server computer networks, when arrival and service rates of nodes depend on amount of unfinished work and number of available tasks on server (see, eg, [2,3,19,2] Typically, an asymptotic analysis of M t /M t /1andM t /M t /c uses differential equations for transition probabilities and asymptotic analysis, as t, for their solutions This type of analysis is similar to an investigation of stability for Markov

2 354 ABRAMOV AND LIPTSER chain and is associated with a verification of stationarity (quasi-stationarity (see [1,4,5, 7 1,21,29] and many others We mention now results related to time-nonhomogeneous stochastic models converging, as t, to time-homogeneous ones Although first results were published more than 3 years ago by Gnedenko and Soloviev [12] and Gnedenko [11], a remarkable progress was achieved not a long time ago (see [13,14,16,23,3 36] In particular, Zeifman developed a number of effective tools, permitting investigate successfully ergodicity conditions for special classes of time-nonhomogeneous birth-and-death processes including M t /M t /1, M t /M t /S and M t /M t /S/ queues (for further discussion, see [14] In the present paper, we give sufficient conditions, under which a time-nonhomogeneous countable Markov chain with the transition intensity matrix (t shares the iting distribution with time-homogeneous ergodic Markov chain with the transition intensity matrix Our setting is heavily related to the above-mentioned settings and the result obtained supplements the results from [14,31,32,35,36] (more detailed comparison is given in section 6 The main difference with known approaches to this problem is that the convergence (t, t is not required In contrast to that we assume the existence of nonnegative λ ij,i j such that (here λ ij (t are entries of (t λ ij <, j i t λ ij (si (λ ij > ds = ( λij (t λ ij 2 dt <, t λ ij (s ds, t >, (11 and create the matrix with entries λ ij,i j and λ ii = j i λ ij We assume that the Markov chain with this transition intensity matrix is ergodic To explain our method with more details, notice that (11 guarantees the absolute continuity of the distribution for (t-markov chain with respect to the distribution for -Markov chain It is also assumed that -Markov chain is ergodic but the geometrical ergodicity is not required We show in theorem 21 (section 2 that the above-mentioned absolute continuity of distributions provides the iting distribution, as t, for (t-markov chain coinciding with the invariant measure of -Markov chain In section 3, we give the proof of theorem 21 In section 4, we show that not only the iting distribution but also other iting functionals are the same as for -Markov chain In section 5, an asymptotic equivalence of (t-markov chain to an ergodic birthand-death process is established

3 LIMITING DISTRIBUTION FOR TIME-NONHOMOGENEOUS MARKOV PROCESS The main result We consider a nonhomogeneous Markov chain Q = (Q t t with the countable set of states S = {, 1,} and the transition intensity matrix (t with entries λ ij (t Suppose that for any pair (i, j with i j there is a nonnegative constant λ ij such that (11 holds true, and introduce Markov chain Q = (Q t t with the set of its states S, and Q = Q, and the transition intensity matrix (see section 1 Our main assumption is that Q is ergodic, ie there is the unique probabilistic measure π on S such that π = and Q t = j Q s = i = π j, s,i,j, (21 where π j are entries of π Theorem 21 Under (11 and (21, 3 The proof of theorem Preinaries P(Q t = j Q s = i = π j, s,i,j (22 Without loss of generality one may assume that the Markov chains Q and Q have paths in the Skorokhod space D = D [, of right continuous having its to the left functions x = (x t t Let ν, ν be the distributions of Q and Q respectively, that is ν, ν are probabilistic measures on (D, G,whereG is the Borel σ -algebra on D Without loss of generality we may assume that G is completed with respect to the measure (ν + ν /2 We shall use in the sequel that ν ν Recall (see, eg, [28] that ν ν provides that ν(a = for any A G, ifν (A = For the verification of ν ν, we apply [17, theorem 24] Following this theorem, ν ν if for all i, j S (a ν (x = i = ν(x = i = ; (b t I(x s = jλ ij (s ds = t I(x s = jλ ij (si (λ ij ds, ν-as; (c with = ( [ P 1 j i ] 2 λ ij (t I(λ ij λ ij I(x t = idt < = 1, ν-as λ ij Notice that for any j i, (c is provided by the condition [ ] 2 λ ij (t 1 I(λ ij λ ij dt < λ ij equivalent to the first part in (11

4 356 ABRAMOV AND LIPTSER Introduce a stochastic basis (D, G,(G t t,ν with the general condition (see, eg, [22], where (G t t is the filtration generated by x Henceforth, E ν and E ν denote the expectations with respect to ν and ν, respectively Set Z(x = dν/dν (x and Z t (x = E ν (Z G t (x We shall use the fact that (Z t (x t is positive uniformly integrable martingale with respect to ν Throughout the paper we use the notation ( for minimum (maximum of two numbers 32 Auxiliary lemma Lemma 31 Under the assumptions of theorem 21, for any s andj S P(Q t = j Q s prob π j Proof With s<s <t, using Markov property, write P(Q t = j Q s = ν(x t = j G s = E ν( ν(x t = j G s x s According to well known formula for the conditional expectation under absolute continuous change of measure: for any integrable random variable α, E ν (α G s = E ν ((Z/Z s α G s we find ( Z(x ν(x t = j G s = E ν Z s (x I(x t = j G s = ν (x t = j x s + E ν ([ Z(x Z s (x 1 ] I(x t = j G s = j x s ν as π j and by ν ν the same convergence holds By (21, ν (x t ν-as too well So, it remains to show that ([ ] Z(x Eν Z s (x 1 I(x t = j G s Notice that ([ ] Z(x E ν E ν Z s (x 1 Z(x E ν Z s (x 1 ( Z(x E ν Z s (x 1 3 ( Z(x = E ν Z s (x 1 3 E ν ( Z(x Z s (x + 1 I I(x t = j G s ( Z(x + E ν Z s (x + 1 I ( Z(x + E ν Z s (x + 1 ( Z(x Z s (x 2 ν prob (31 s ( Z(x Z s (x > 2

5 LIMITING DISTRIBUTION FOR TIME-NONHOMOGENEOUS MARKOV PROCESS 357 As was mentioned above Z t (x is the positive uniformly integrable ν -martingale Hence s Z s (x = Z(x, ν -as Consequently, by Lebesgue dominated theorem ( Z(x s Eν Z s (x 1 3 = (32 Now, it remains to show that ( ( ( Z(x Z(x Z(x s Eν Z s (x + 1 = s Eν Z s (x + 1 I Z s (x 2 Since Z s (x Z(x,s, by the Lebesgue dominated theorem the right hand side of the above equality is equal to 2 At the same time for any s we have E ((Z(x/Z s (x G s = 1andsoforanys it holds E (Z(x/Z s (x + 1 = 2 Thus, ([ ] Z(x Eν Z s (x 1 ν I(x t = j G prob s s 33 Final part of proof By lemma 31 we have P(Q t = j Q s = ii(q s = i prob π j Hence, for any i S j= P(Q t = j Q s = ii(q s = i prob ji(q s = i (33 and the statement of theorem 21 follows 4 Asymptotic equivalence for other functionals Denote h(x t = I(x t = j Theorem 21 guarantees the asymptotic equivalence Eν( ( h(x t G s = E ν h(xt G s An analysis of the proof of theorem 21 shows that the same type of asymptotic equivalence holds for any bounded functional h(x [t, of argument x [t, ={x u,u t} provided that E ν (h(x [t, G s exists, that is under the assumptions of theorem 21 Eν( h(x [t, G s = E ν ( h(x[t, G s (41

6 358 ABRAMOV AND LIPTSER 5 Asymptotic equivalence to birth-and-death process Let λ (t λ (t µ 1 (t (λ 1 (t + µ 1 (t λ 1 (t (t = µ 2 (t (λ 2 (t + µ 2 (t λ 2 (t µ 3 (t (λ 3 (t + µ 3 (t and Assume that there exist positive numbers λ j,µ j s such that n n=1 j=1 λ j 1 µ j < (51 [ ( λj (t λ j 2 + ( µj (t µ j 2 ] dt <, j (52 We mention here that (52 does not provide λ j (t λ j,µ j (t µ j, Introduce the matrix λ λ µ 1 (λ 1 + µ 1 λ 1 = µ 2 (λ 2 + µ 2 λ 2 µ 3 (λ 3 + µ 3 and notice that Markov chain with the transition intensity matrix generates the birthand-death process It is well known (see, eg, [18, chapter 7, section 5] that under (51 the birth-anddeath process is ergodic with the unique stationary distribution on S: π = 1 + n=1 1 n n j=1 (λ j 1/µ j, π n = π j=1 λ j 1 µ j, n = 1, 2, (53 By theorem 21, the Markov chain with transition intensity matrix (t possesses the same stationary distribution It is known that the sojourn time T i in state i for the birth-and-death process is exponentially distributed P(T i x = 1 e (λ i+µ i x Under (51 and (52, for Markov chain with the transition intensity (t we have the following Let T i (t = v i (t t, where v i (t = inf{s t: Q s i, Q t = i}

7 LIMITING DISTRIBUTION FOR TIME-NONHOMOGENEOUS MARKOV PROCESS 359 Then, applying the result from section 4, we obtain P ( T i (t x = 1 e (λ i+µ i x 6 Discussion We consider here M t /M t /1 model Let A t and D t be independent and timenonhomogeneous Poisson processes with positive rates λ(t and µ(t respectively Let Q be a random variable, independent of A t and D t, taking values in S ={, 1,} We define the queue-length process Q = (Q t t in the M t /M t /1 as follows (notice that jumps of A t and D t are disjoint and so Q t S Q t = Q + A t t I(Q s > dd s Let A t and Dt be independent and homogeneous Poisson processes with positive rates λ and µ, letq = Q be independent of A t and Dt,andletQ = (Q t t be the queue-length process in the M/M/1 queue with parameters λ and µ for arrival and service of customers, respectively, The queue-length process Q t is defined as by Q t = Q + A t λ<µ t I ( Q s > dd s By theorem 21 the existence of the iting distribution for M t /M t /1 is provided [ ( λ λ(t 2 + ( µ µ(t 2 ] dt < (61 On the other hand, it is known from [14,31,32,35] that the existence of the iting distribution is provided by [ λ λ(t + µ µ(t ] = (62 Generally, (61 (62, and so (61 and (62 supplement each other If λ(t and µ(t are uniformly continuous on [, functions, then (61 (62, that is (62 is weaker than (61 Notice also that (62 (61, say, under additional condition: for small positive ε and t large enough λ λ(t + µ µ(t =O(t (1/2+ε It would be noted that in [36], it is studied an ergodicity problem, in the uniform operator topology, of time-nonhomogeneous Markov chains with transition intensity ma-

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