Fluid Models with Jumps

Transcription

1 Flui Moels with Jumps Elena I. Tzenova, Ivo J.B.F. Aan, Viyahar G. Kulkarni August 2, 2004 Abstract In this paper we stuy a general stochastic flui moel with a single infinite capacity buffer, where the buffer content can change continuously as well as by instantaneous upwar jumps. The continuous as well as the instantaneous change is moulate by an external environment process moelle as a finite state continuous time Markov chain. The Laplace-Stieltjes transform of the steay-state joint istribution of the buffer content an the state of the environment is etermine explicitly in terms of the solutions of a generalize eigenvalue problem. The methoology is illustrate by several well-known queueing problems. 1 Introuction A stochastic flui queueing system escribes the input-output flow of a flui in a storage evice, calle a buffer. The rates at which the flui enters an leaves the buffer epen on a ranom environment process that is usually chosen to be an irreucible CTMC. The stuy of flui flow moels is motivate by their various applications. One of these are the real-worl systems that eal with the processing of continuous entities such as the ones use in the petroleum an chemical inustries. Flui flow moels also provie an important tool for the performance analysis of high-spee ata networks, or large-scale prouction systems where a large number of relatively small jobs are processe. Flui flow moels are also use as moels of the asymptotic behavior of queues in heavy traffic. Most of the classical research on stochastic flui systems in the area of telecommunications is base on the work of Anick, Mitra an Sonhi [2] which is an extension of the pioneering work of Kosten [4]. They stuy a moel with several ientical an inepenent Markov on-off sources that transmit flui to an infinite-capacity buffer. The flui is then Department of Mathematics, Tulane University, New Orleans, LA Department of Mathematics an Computer Science, Einhoven University of Technology, P.O. Box 513, 5600 MB Einhoven, The Netherlans Department of Operations Research, University of North Carolina, CB 3180, Chapel Hill, N.C

2 processe at a fixe rate. The limiting istribution of the buffer content process is compute as a solution of a set of orinary ifferential equations. The ientical input sources facilitate the analysis of the ifferential equations an the main result provies the system s eigenvalues in explicit form. In [8], [9] Mitra generalizes this moel by introucing multiple on-off switching. Most of the work in flui queues eals with the steay-state istribution of the buffer content. See the survey paper by Kulkarni [5] for an extensive overview of the research in this area. In this paper we consier a moification of the classical flui moel, where in aition to the continuous changes, the buffer content can have instantaneous upwar jumps. The rate of the continuous change an the size of the jumps both epen on the state of the environment. The aim of this paper is to etermine the Laplace-Stieltjes transform of the steay-state joint istribution of the buffer content an state of the environment. This problem can be reuce to that of fining the bounary probabilities (of an empty buffer). We show that the bounary probabilities can be explicitly etermine in terms of the solutions of a generalize eigenvalue problem. The evelope technique is illustrate by some simple example of well-known queueing problems. Flui moels with upwar, respectively ownwar, jumps are natural continuous analogues of the M/G/1-type an G/M/1-type Markov chains stuie by Neuts [10, 11]. Recently matrix-analytical results for the iscrete problem have been extene to the continuous case. Takaa [?] stuies the moel with upwar jumps, an takes the following approach to fin the bounary probabilities: instea of using the solutions of an eigenvalue problem, Takaa first etermines the funamental matrix, similar to the one for the M/G/1-type Markov chain, an then obtains the bounary probabilities, up to a multiplicative constant, as the steay-state istribution of the embee Markov chain at visits to the bounary states. Sengupta [12] an Miyazawa an Takaa [?] stuy the flui moel with ownwar jumps; they show that the steay-state joint istribution of the buffer content an state of the environment has a matrix-exponential form, where the exponent matrix is the solution of a non-linear matrix integral equation. 2 The Moel Consier a general stochastic flui moel with a single infinite capacity buffer where the buffer content X(t) can increase continuously as well as by instantaneous jumps. The change of the flui in the buffer epens on the state of an external ranom environment process {I(t), t 0} which is taken to be a stochastic process with a finite state space S = {1,..., N}. While I(t) = i the buffer content increases continuously with rate r i (, ). The process {I(t), t 0} jumps to any state j S (not necessarily ifferent from i) with probability p ij. In state i S it will make a jump after an exponential amount of time with mean 1/q i. When the I(t) process jumps from state i to state j the amount of flui in the buffer can increase by a lump-sum non-negative ranom amount with a given 2

3 c..f. G ij (y), y 0 an mean m ij. Thus, the bivariate Markov process {(X(t), I(t)), t 0} can jump from state (x, i) to state (x + y, j) with rate q i p ij G ij (y), y 0. Let R enote the iagonal N N matrix with the net input rates r i along the iagonal an also efine R := iag[r 1,..., r N ], Q ij (x) := q i p ij G ij (x), x 0, i, j S, i j, (2.1) Q ii (x) := q i p ii G ii (x) q i, x 0, i S. (2.2) Note that {I(t), t 0} itself is a CTMC on S with rate matrix Q = Q( ). We assume that {I(t), t 0} is irreucible. Let π i := lim P (I(t) = i) be the limiting istribution of the I(t) t process. The following results states that the system is stable if the mean net input rate in steay state is negative (cf. Loynes [?]). Theorem 2.1 The system is stable if π i (r i + q i p ij m ij ) < 0. (2.3) j=1 Proof: Consier the sample chain at jumps of the I(t) process to state 1. Let X n enote the buffer content at the n-th jump to state 1. Then it is reaily verifie that lim E(X n+1 X n = x) = x + 1 x π 1 q 1 π i (r i + q i p ij m ij ). j=1 Hence, by virtue of (2.3), there exist positive constants ɛ, b, K such that for all x 0, E(X n+1 X n = x) x ɛ + b1 [x K], where 1 [ ] is the inicator function. From Foster s criterion (see e.g. Meyn an Tweeie [?]) we can now conclue that the sample chain is stable. Since the times elapsing between successive jumps to state 1 are ii with finite mean, the result follows for the original chain. Define the matrix Γ = [Γ ij ] by Γ ij = q i p ij m ij. Then the stability conition (2.3) can be written in matrix form as follows π (R + Γ) e < 0. We shall assume from now on that this stability conition hols. Now, enote F i (t, x) := P (X(t) x, I(t) = i), x, t 0, i S, 3

4 an F i (x) := lim t P (X(t) x, I(t) = i), x 0, i S. The next theorem shows that F i (x) is ifferentiable for x > 0 an gives the ifferential equations satisfie by F i (x). First we introuce the following notation: F (x) := [F 1 (x),..., F N (x)], [ F x := F1 x,..., F ] N, x S := {i S : r(i) < 0}, N := S, S 0 := {i S : r(i) = 0}, N 0 := S 0, S + := {i S : r(i) > 0}, N + := S +. Theorem 2.2 The limiting istribution F (x) satisfies The bounary conitions are given by where Q.j (0) is the j-th column of Q(0). F R = F Q(x). (2.4) x F i (0) = 0, i S +, (2.5) F (0)Q.j (0) = 0, j S 0, (2.6) Proof: First, consier F j (t, x), x > 0, j S an conition on a small time interval of length h > 0 so that F j (t, x) = x ri h z=0 P (X(t h) x r i h z, I(t h) = i)q i p ij hg ij (z) + (1 q j h)p (X(t h) x r j h, I(t h) = j) + o(h) x ri h = h F i (t h, x r i h z)q i p ij G ij (z) + (1 q j h)f j (t h, x r j h) + o(h). z=0 Now let t an rearrange the last equation so that F j (x) F j (x r j h) h x ri h = F i (x r i h z)q i p ij G ij (z) q j F j (x r j h) + o(h) z=0 h (2.7) From the efinition of Q(x) in (2.1) we have Q ij (x) = q i p ij G ij (x), x 0, i, j S, i j. 4

5 an from (2.2), Q ii (x) = q i p ii G ii (x), x > 0, i S. Since Q ii (0 ) = G ii (0 ) = 0 we obtain for the jump of Q ii (x) at x = 0, Q ii (0) = Q ii (0) = q i p ii G ii (0) q i, i S. Now, we can write Eq. (??) in the following nice form: F j (x) F j (x r j h) h x ri h = F i (x r i h z)q ij (z) + o(h) z=0 h. After we let h 0 we get r j F j (x) x x = F i (x z)q ij (z) = F i Q ij (x), (2.8) z=0 which in matrix form becomes Eq. (2.7), that is, F x R = F Q(x). It remains to establish the bounary conitions. From the efinition of F i (x) we have F i (0) = lim t P (X(t) = 0, I(t) = i), i S. Therefore, for states i S + with positive net input rates the long-run probabilities F i (0) are zero. Thus, the first set of bounary conitions (2.8) is given by F i (0) = 0, i S +. To get the secon set of bounary conitions (2.9) we can again apply conitioning on a small time interval of length h > 0, for x = 0, j S 0, as above to obtain ( ri h) + z=0 F i (t h, ( r i h) + z)q i p ij hg ij (z) + (1 q j h)f j (t h, 0) = F j (t, 0), j S 0, where ( ) + = max(, 0). After letting t an using the notation Q ij (z) we get ( ri h) + z=0 F i (( r i h) + z)q ij (z) = 0, j S 0. Now, as h 0 the bounary conitions (2.9) are obtaine F i (0)Q ij (0) = 0, j S 0. 5

6 Remark: Equations (2.7)-(2.9) have a simple rate interpretation. For example, if r j > 0, then the rate out of the set of states {(y, j), 0 y x} is equal to an the rate into that set, r j F j (x) x Equating the two rates yiels Eq. (2.12). F j Q jj (x) F i Q ij (x). i j We solve the ifferential equations (2.7) of the above theorem using Laplace Stieltjes Transforms (LST). By Eqs. (2.1) an (2.2), The LST of Q(x) is given by where the LSTs are efine as usual, i.e. Q ij (s) := 0 Q ij (s) = q i p ij Gij (s), i, j S, i j, (2.9) Q ii (s) = q i p ii Gii (s) q i, i S, (2.10) e sx Q ij (x), Gij (s) := 0 e sx G ij (x), i, j S. Note that one nees to account for the jump of Q ii (x) at x = 0 in eriving Eq. (2.14). The Eqs. (2.13) an (2.14) imply that Q(0) = Q is the generator matrix of the CTMC {I(t), t 0}. Taking the LST of both sies of Eq. (2.7) we get F (s)(sr Q(s)) = sf (0)R. (2.11) Thus, in orer to fin the LST F (s) we nee to know F (0) = [F 1 (0),..., F N (0)]. From the bounary conitions (2.8) we have that N + components of F (0) are zero an N 0 components can be expresse in terms of the remaining N from the secon set of bounary conitions (2.9). The next theorem is use to etermine the remaining N = N N + N 0 components of F (0) (cf. Theorem 5 in Loynes [6]). Theorem 2.3 Assume Conition (2.3) hols. Then the generalize eigenvalue problem (sr Q(s))φ = 0 (2.12) has exactly N solutions (s 1, φ 1 ),..., (s N, φ N ), with s 1 = 0, Re(s i ) > 0, i = 2,..., N an φ i 0. Furthermore, these zeros s 1,..., s N lie on or insie the circle in the complex plane with center at λ = max q i an raius λ. i:r i 0 r i 6

7 Proof: is given in the Appenix. The next theorem is the main result of this paper. It summarizes the entire set of equations satisfie by F (0). Theorem 2.4 Assume Conition (2.3) hols. Then the LST row vector F (s) satisfies F (s)(sr Q(s)) = sf (0)R, (2.13) where the unknowns F (0) = [F 1 (0), F 2 (0),..., F N (0)] are given by the unique solution to the N equations F i (0) = 0, i S +, (2.14) F (0)Q.,i (0) = 0, i S 0, F (0)Rφ i = 0, for i = 2,..., N, (2.15) F (0)Re = π(r + Γ)e. (2.16) Proof: The equation for F (s) is as erive above, Eq. (2.15). Also, recall the first two sets of equations for F i (0) when i S + or i S 0, given in Theorem 2.1, Eqs. (2.8) an (2.9). Since F (s) is analytic for Re(s) 0, it must be the case that for every i, F (s i )(s i R Q(s i ))φ i = s i F (0)Rφ i = 0. However, for s 1 = 0 the last equation is trivially satisfie an we nee one aitional conition to etermine F (0). Thus, we get Eqs. (2.20). Now, let M(s) := sr Q(s) an M (s) be the erivative of M(s). To erive the normalization equation (2.21) first recall that from (2.17) we have F (s)m(s) = sf (0)R. (2.17) After ifferentiating equation (2.23) an setting s = 0 we get F (0)M (0) + F (0)M(0) = F (0)R. Post-multiplying by e an noting that M(0)e = Q(0)e = 0, yiels F (0)Re = F (0)M (0)e. Since F (0) = F ( ) = π an M (0) = R + Γ, we get Eq. (2.21). The uniqueness of the solution follows from the assume stability an by following similar arguments as in the proof of Theorem 3 of Gail et al. [?]. Remark: Theorem 2.3 characterizes the LST F (s) as the solution to Eq. (2.17), where the bounary vector F (0) is etermine by the solutions to the generalize eigenvalue problem 2.16 through the Eqs. (??)-(2.21). The equivalent result to Theorem 2.3 is given in Takaa [?], Proposition 2.1, where, as alreay mentione in the Introuction, the bounary vector F (0) is etermine through the funamental matrix. 7

8 3 Examples We illustrate the above methoology with the help of several examples. 3.1 The M G 1 Queue. Consier a stanar M G 1 queue with arrival rate λ an service time istribution G with mean τ. Let X(t) be the work content in this system at time t. The {X(t), t 0} process ecreases at rate 1 an jumps up by a ranom amount with cf G whenever an arrival occurs. If X(t) becomes zero, it stays zero until an upwar jump occurs. Thus we can moel this as a flui process with jumps with a single state environment process. The parameters are as follows: S = {1}, q 1 = λ, p 1,1 = 1, G 1,1 = G, m 1,1 = τ, r 1 = 1. Hence π 1 = 1, an the conition of stability (2.3) reuces to 1 + λτ < 0, or ρ = λτ < 1. This is the stanar conition of stability for the M G 1 queue. We get Q(s) = λ( G(s) 1), an Equation (2.17) reuces to the scalar equation F (s)(s λ(1 G(s))) = sf (0). There is only one bounary conition, given by Eq. (2.21). It reuces to F (0) = 1 λτ = 1 ρ. Using that we get s(1 ρ) F (s) = s λ(1 G(s)). This matches the LST of the queueing time in an M G 1 queue, as it must. 3.2 The BMAP G 1 Queue. Consier a single server queue whose arrival process is given by a BMAP efine by the sequence {D k, k 0}. Here D 0 has negative iagonal elements an nonnegative off-iagonal elements, D k, k 1, are nonnegative, an D, efine as D = D k, k=0 is the generator an irreucible CTMC on state space S = {1, 2,..., N}. Transitions accoring to D k correspon to batch arrivals of size k. The service times are ii ranom variables with cf G an mean τ. Let I(t) be the phase of the arrival process at time t. It is a CTMC on S with generator matrix D. Let π be the limiting istribution of I(t). Let X(t) be the work content in the system at time t. The {X(t), t 0} process ecreases at rate 1 an jumps up by a ranom amount with cf G k whenever a batch arrival of size k occurs. If X(t) becomes zero, it stays zero until an upwar jump occurs. Thus we can moel the work content process as a flui process with jumps with the following parameters: The state space of the environment process (the phase of the arrival process) is S = {1, 2,..., N}. We get Q(s) = D Gk k (s) = D( G(s)), k=0 8

9 where D(z) = D k z k, z 1. k=0 We also have R = I, since the net rate is 1 in every state. The conition of stability is π kd k eτ < 1. k=1 Since S = S, Theorem 2.2 implies that there are N (eigenvalue, eigenvector) pairs (s i, φ i ) (i = 1, 2,..., N) satisfying (si + D( G(s)))φ = 0, with s 1 = 0 an Re(s i ) > 0 for i = 2, 3,..., N. Finally, Theorem 2.3 yiels the following result for the LST of F : F (s) = sf (0)(sI + D( G(s))) 1, (3.18) where the unknown vector F (0) is obtaine from the N equations: F (0)φ i = 0, i = 2, 3,..., N. F (0)e = 1 π kd k eτ. k=1 This matches with the known results; see Eq. (44) in Lucantoni [7]. 3.3 The P H G 1 Queue. The P H arrival process with parameters (α, T ) is a special case of the BMAP. Here D 0 = T, D 1 = T eα an D k = 0 for all k > 1, where T = [t i,j ] i,j=1,2,...,n is an invertible generator matrix with non-positive row sums of an irreucible CTMC on state space S = {1, 2,..., N}, an α = [α 1, α 2,..., α N ] is a non-negative vector whose elements a up to one. Eq. (3.25) for the LST of F now simplifies to F (s) = sf (0)(sI + T T eα G(s)) 1, which matches with Eq. (5.2.4), p. 257 of Neuts [10]. 3.4 The M G 1 Queue with an Up-Down Server. Consier the queueing system of the first example, with the following moification: The server can be up or own. The up times are Exp(β) ranom variables, an the own times are Exp(α) ranom variables. Let I(t) be the state of the server (1 if up, an 0 if own) at time t. Thus {I(t), t 0} is a CTMC. Let X(t) be the work content at time t in this system. We moel {(X(t), I(t)), t 0} as a flui queue with jumps with the following parameters: q 0 = λ + α, q 1 = λ + β 9

10 so that an p 01 = α α + λ, p 00 = G 01 (x) = G 10 (x) = 1, λ α + λ, p 10 = β λ + β, p 11 = λ λ + β G 00 (x) = G 11 (x) = G(x) [ ] (α + λ(1 G(x))) α Q(x) =, (3.19) β (β + λ(1 G(x))) R = This implies that S 0 = {0} an S = {1}. The stability conition is given by We have ρ = λτ < [ 0 1 ]. α α + β = π 1. [ ] (α + λ(1 G(s))) α Q(s) = β (β + λ(1 G(s))). (3.20) Theorem 2.3 yiels the following result for the LST of F (x) = [F 0 (x), F 1 (x)]: F (s) = sf (0)(sR Q(s)) 1. The bounary conitions yiel the following explicit expression for F (0) = [F 0 (0), F 1 (0)]: F 0 (0) = This can be simplifie to get F 0 (s) = F 1 (s) = β ( ) α α + λ α + β ρ, F 1 (0) = α α + β ρ. sβ(π 1 ρ) αβ (α + λ(1 G(s)))(β + λ(1 G(s)) s), s(π 1 ρ)(α + λ(1 G(s)) αβ (α + λ(1 G(s)))(β + λ(1 G(s)) s). This matches the results in Gaver [3]. As a special case, when β = 0, the server is permanently up, an we get the stanar queue of the first example. The results match for this special case. 10

11 3.5 An Inventory Moel Consier the following inventory moel. Deman occurs continuously at rate µ. A machine prouces continuously at rate r > µ. It is subject to failures an repairs; the up times are Exp(β) ranom variables an the own times Exp(α) ranom variables. In aition, an external supplier elivers the prouct accoring to a compoun Poisson process with batch arrival rate λ an batch size istribution G with mean τ <. All unsatisfie eman is lost. Let X(t) be the inventory at time t an let I(t) be the state of the machine at time t (1 if up, an 0 if own). Then {(X(t), I(t)), t 0} is a flui moel with jumps with r 0 = µ < 0 an r 1 = r µ > 0. Clearly, S = {0} an S + = {1}; so in this example we have a non-empty S +. The parameters q i, p ij, G ij (x), i, j = 0, 1, are the same as in Example??. Hence the matrices Q(x) an Q(s) are as in Eq. (??) an Eq. (??) respectively. The system is stable when λτ < µ α α + β r. By Theorem 2.3, F 1 (0) = 0 from Eq.(??) an F 0 (0) is etermine from Eq.(2.21) an the LST is where F 0 (0) = µ α r λτ α+β, µ F (s) = sf (0)(sR Q(s)) 1 = sf 0(0) (s(r µ) + β + λ(1 G(s)), α), = et(sr Q(s)) = ( sµ + α + λ(1 G(s)))(s(r µ) + β + λ(1 G(s))) αβ. 4 Appenix Proof of Theorem 2.2: We follow the metho use in a similar proof in [1]. Note that it follows from Eqs. (2.13) an (2.14) that we can rewrite the matrix Q(s) in the form Q(s) = Q(s) Q, where Q(s) := [q i p ij Gij (s)], an Q := iag[q 1,..., q N ]. Then Eq. (2.16) has a solution with φ 0 if et( Q(s) Q sr) = 0, which is equivalent to et(q 1 Q(s) sq 1 R I) = 0. 11

12 We first assume that for some ɛ > 0 the transformations G ij (s) are analytic for all s with Re(s) > ɛ. For example, this hols for istributions with an exponential tail or istributions with finite support. Let an let ( λ := max i:r i 0 q i r i ), C δ := {s : s λ = λ + δ}, enote the circle with center at λ an raius λ + δ, where 0 < δ < ɛ. Next, we show that for 0 u 1 an small δ > 0, et(uq 1 Q(s) sq 1 R I) 0, s C δ. (4.21) First, in the case s C δ with Re(s) 0 we prove that the matrix uq 1 Q(s) sq 1 R I is iagonally ominant (with strict ominance in at least one row). This, plus the fact that P = [p ij ] is irreucible, imply (see [13]) that it has a non-zero eterminant. 1. Let i : r i 0 an enote by λ i := q i r i so that λ := max i:r i 0 λ i. Clearly, up ii Gii (s) sr i q i 1 = up ii Gii (s) + s s 1 1 up ii Gii (0). λ i λ i i : r i < 0 or equivalently i : λ i > 0 Then we have s 1 = s λ i s λ = λ i λ i λ i an therefore λ + δ λ i λ i + δ λ i = 1 + δ λ i, up ii Gii (s) sr i δ up ii Gii (0) > u q i λ G ii (0) up ii Gii (0) = i j i j i up ij Gij (s). up ij i : r i > 0 or equivalently i : λ i < 0 Then it is clear that for Re(s) 0 an s C δ s 1 λ > 1, i an therefore up ii Gii (s) sr i 1 > 1 up ii Gii (0) u q G ii (0) up ii Gii (0) = i j i j i up ij Gij (s). up ij 12

13 2. Let i : r i = 0. Then up ii Gii (s) 1 1 up ii Gii (0) u G ii (0) up ii Gii (0) = up ij up ij Gij (s). j i j i Note that the stability conition (2.3) implies that at least one of the rates r i < 0. Thus we have strict ominance in at least one row. Further, the arguments above apply to any circle C δ with δ > 0. This implies that the eterminant in (4.31) has no zeros in the right half plane outsie C 0. Next, consier the case s C δ with Re(s) < 0. Note that the eterminant of (4.31) is nonzero if an only if 0 is not an eigenvalue. Therefore, next we stuy the eigenvalues of Q(s) sq 1 R I in a neighborhoo of s = 0. If we write uq 1 uq 1 Q(s) sq 1 R I = P I sq 1 R + (u 1)Q 1 Q(s) + Q 1 Q(s) P, we can see that the above matrix is a perturbation of P I when (s, u) is close to (0, 1). Since P is assume to be irreucible, P I has a simple eigenvalue 0. Thus, in a neighborhoo of (0, 1), there exist ifferentiable x(s, u) an µ(s, u) such that (uq 1 Q(s) sq 1 R I)x(s, u) = µ(s, u)x(s, u), µ(0, 1) = 0, x(0, 1) = e Differentiating the last equation with respect to s an setting s = 0, u = 1 in the result gives the following: ( Q 1 Q(0) s ) Q 1 R e + (P I) s x(0, 1) = µ(0, 1)e, s After pre-multiplying this equation by πq, where π := [π 1,..., π N ], π i = lim P (I(t) = i) t we get ( ) Q(0) π R e + πq (P I) s s x(0, 1) = s µ(0, 1)πQ e. Note that π is the solution to πq (P I) = 0, an also from the efinition of Q(s) above we have Q(s) s = [ q i p ij G ij (s) s ], an therefore Q(0) s = [ q i p ij m ij ] = Q P M, 13

14 where P M enotes the Haamar matrix multiplication. Hence we have π ( Q P M R) e = s µ(0, 1)πQ e. (4.22) In a similar way ifferentiation with respect to u leas to the following: an since πq P e = u µ(0, 1)πQ e, πq P e = π i q i p ij = π j q j = πq e, j=1 j=1 we have µ(0, 1) = 1. u Therefore, by Taylor series expansion of µ(s, u) in a neighbourhoo of (0, 1) from the last equation an Eq. (4.42) it follows µ(s, u) = s π ( Q P M R) e πq e From the stability conition (2.3) we have π ( Q P M R) e > 0, + u 1 + o( s 2 + (u 1) 2 ). so we can conclue that µ(s, u) 0 for small δ, s C δ with Re(s) < 0, an u close to 1, say 1 ˆδ u 1. For 0 u < 1 ˆδ, it can be shown that uq 1 Q(s) sq 1 R I is iagonally ominant (with strict ominance in at least one row) for s C δ with Re(s) < 0, provie that δ is taken small enough so that For i : r i < 0 we have as before 1 δ λ i > (1 ˆδ)max i,j S G ij ( δ). up ii Gii (s) sr i q i δ λ i up ii Gii (s) > 1 δ λ i up ii Gii (s) > For i : r i 0 it is obvious that > (1 ˆδ) max i,j S > u max i,j S G ij ( δ) up ii Gii (s) > G ij ( δ) up ii max i,j S G ij ( δ) up ij Gij (s). j i up ii Gii (s) sr i q i 1 1 up ii Gii (s) > 1 δ λ i up ii Gii (s), 14

15 an the rest of the argument is the same as above. This completes the proof of (4.31). Let f(u) enote the number of zeros of et(uq 1 Q(s) sq 1 R I) insie the circle C δ. Then from the Cauchy Theorem of Complex Analysis we have f(u) = 1 2πi C δ It is clear that f(0) = N, since s et(uq 1 Q(s) sq 1 R I) et(uq 1 Q(s) sq 1 s. R I) N ( ) et( sq 1 R I) = sri + 1, q i an therefore the zeros are s i = q i r i, r i 0 an exactly N of them have Re(s) 0 an thus are insie the circle C δ. Since f(u) is an integer-value continuous function on [0, 1] it follows that it is constant. Hence, f(1) = N. As δ 0, we can conclue that et(sr Q(s)) has exactly N zeros insie or on C 0. It is clear that s = 0 satisfies et(sr Q(s)) = 0. It can be shown that s = 0 is a simple root of this equation using the irreucibility of P an the stability conition (2.3). The arguments are the same as in [1] an therefore we skip them. To finally complete the proof of Theorem 2.2 we have to remove the initial assumption that for some ɛ > 0 the transforms G ij (s) are analytic for all s with Re(s) > ɛ. To this en, first consier the truncate istributions G K ij (x) efine as G K ij (x) = G ij (x) for 0 x < K an G K ij (x) = 1 for x K. Then Theorem 2.2 hols for the istributions G K ij (x); by letting K ten to infinity, the result also follows for the original istributions. References [1] I.J.B.F. Aan an V. Kulkarni (2003) Single server queue with Markov epenent inter-arrival an service times. Queueing Systems 45, [2] D. Anick, D. Mitra an M.M. Sonhi (1982) Stochastic theory of a ata hanling system with multiple sources. Bell Syst. Tech. J. 61, [3] H.R. Gail, S.L. Hantler, B.A. Taylor (1992) On a preemptive Markovian queue with multiple servers an two priority classes. Math. Oper. Res. 17, [4] D.P. Gaver (1962) A waiting line with interrupte service, incluing priorities. J. R. Statist. Soc., Series B, 24(1),

16 [5] L. Kosten (1974) Stochastic theory of a multi-entry buffer, part 1. Delft Progress Report, [6] V.G. Kulkarni (1997) Flui moels for single buffer systems. Frontiers in Queueing; Moels an Applications in Science an Engineering, [7] R.M. Loynes (1962) The stability of a queue with non-inepenent inter-arrival an service times. Proc. Camb. Philos. Soc. 58, [8] R.M. Loynes (1962) Stationary waiting time istributions for single-server queues. Ann. Math. Statist. 33, [9] D.M. Lucantoni (1991) New results on the single server queue with a batch Markovian arrival process. Stochastic Moels 7(1), [10] S.P. Meyn, R. Tweeie (1994) Markov Chains an Stochastic Stability. Springer- Verlag, New York. [11] D. Mitra (1988) Stochastic flui moels. Performance 87, Elsevier Science Publishers, North Hollan, [12] D. Mitra (1988) Stochastic theory of a flui moel of proucers an consumers couple by a buffer. Av. App. Prob. 20, [13] M. Miyazawa, H. Takaa (2002) A matrix exponential form for hitting probabilities an its application to a Markov-moulate flui queue with ownwar jumps. J. Appl. Prob. 39, [14] M. Neuts (1989) Structure stochastic matrices of M G 1 type an their applications. Marcel Dekker, New York. [15] M. Neuts (1981) Matrix-Geometric Solutions in Stochastic Moels. An Algorithmic Approach. Dover Publications, Inc., New York. [16] B. Sengupta (1989) Markov processes whose steay state istribution is matrixexponential with an application to the GI/P H/1 queue. Av. Appl. Prob. 21(1), [17] Jos H. A. De Smit (1983) The queue GI/M/s with customers of ifferent types or the queue GI/H m /s. Av. Appl. Prob. 15, [18] H. Takaa (2001) Markov moulate flui queues with batch flui arrivals. J. Operat. Res. Soc. Japan 44, [19] M. Miyazawa, H. Takaa (2002) A matrix exponential form for hittinh probabilities an its application to a Markov-moulate flui queue with ownwar jumps. J. Appl. Prob. 39,