Existence Condition for the Diffusion Approximations of Multiclass Priority Queueing Networks Hong Chen 1 Faculty of Commerce and Business Administration University of British Columbia, Vancouver, Canada
Heng Qing Ye 2 Department of Decision Sciences, Faculty of Business Administration National University of Singapore, Singapore
Abstract In this paper, we extend the work of Chen and Zhang (2000b) and establish a new sufficient condition for the existence of the (conventional) diffusion approximation for multiclass queueing networks under priority service disciplines. This sufficient condition relates to the weak stability of the fluid networks and the stability of the high priority classes of the fluid networks that correspond to the queueing networks under consideration. Using this sufficient condition, we prove the existence of the diffusion approximation for the last-bufferfirst-served reentrant lines. We also study a three-station network example, and observe that the diffusion approximation may not exist, even if the “proposed” limiting semimartingale reflected Brownian motion (SRBM) exists.
Keywords: multiclass queueing network, priority service discipline, diffusion approximation, fluid approximation, heavy traffic, semimartingale reflecting Brownian motion. AMS 1991 Subject classifications: Primary 60F17, 60K25, 60G17; Secondary 60J70, 90B10, 90B22.
Current version: June 2000
1
Supported in part by a grant from NSERC (Canada) and a grant from RGC (Hong Kong). This work was done while the author was a PhD Student in the Hong Kong University of Science and Technology. 2
1
Introduction
We consider a general multiclass queueing network under a preemptive resume priority service discipline. The network consists of J single server (service) stations, each with an infinite buffer (waiting room). There are K (≥ J) classes of jobs, indexed by k = 1, ..., K. While each station may serve more than one class of jobs, each class is served at a specific station. After receiving the service, the job is routed to another station or leaves the system following a Markovian routing. The external arrival processes and the service processes are renewal processes. Each class is assigned a priority. When there is more than one class of jobs at a station, the server at the station serves the job with the highest priority; within a class, jobs are served in the order of arrival. The server of each station is kept busy as long as there are jobs at that station. Let Qk (t) indicate the number of class k jobs in the network at time t, k = 1, ..., K, and let Q(t) = (Qk (t)). The diffusion approximation addresses the weak convergence of the scaled K-dimensional queue ˆ n as n → ∞ under a heavy traffic condition, where length process Q ˆ n (t) = √1 Q(nt). Q n (Roughly speaking, the heavy traffic condition is that the traffic intensity at each station equals one.) Since it is known that the above diffusion approximation may not always exist (see Chen and Zhang (1998)), sufficient conditions are sought for the existence of the diffusion approximation. It turns out that these conditions are related to the behavior of a corresponding fluid network. The fluid network corresponds to the queueing network, consisting of J stations, with K classes ¯ k (t) indicate the class k of (discrete) jobs replaced by K classes of (continuous) fluids. Let Q ¯ ¯ fluid level at time t, k = 1, ..., K, and let Q(t) = (Qk (t)). Let L and H be a partition of the set of all classes (or more precisely indices of classes), with L being the set of all classes with the lowest priority at each station. One of the key conditions in our sufficient condition is that there ¯ ¯ exists a finite time τ ≥ 0 such that for any Q(t) with a unit initial fluid level (i.e., ||Q(0)|| = 1), ¯ QH (t) = 0 for all t ≥ τ . In short, the fluid level of higher priority classes reaches zero in a finite time and then remains in zero; this condition shall be referred to as the SHP -condition, short for the stability of higher priority classes. On the other hand, under the heavy traffic condition, if the weak convergence, ˆn ⇒ Q ˆ as n → ∞, Q ˆ then it is necessary that Q ˆ H ≡ 0 and Q ˆ L is a semimartingale holds with a continuous limit Q, reflected Brownian motion (SRBM) (see Theorem 6 of Chen and Zhang (2000b)). A necessary ˜L condition for the SRBM to be well-defined is that a corresponding J × J reflection matrix R ˜ L depends only on the priority discipline and the to be completely-S. (We note that matrix R first-order parameters of the queueing network, whose precise definition shall be given later. A square matrix A is said to be an S-matrix if there exists a vector x ≥ 0 such that Ax > 0, and it is said to be completely-S if all of its principal submatrices are S-matrices.) The sufficient condition we establish for the existence of the diffusion approximation is that ˜ L is completely-S and the SHP -condition holds. This sufficient condition improves the matrix R the sufficient condition given by Chen and Zhang (2000b), which instead of the SHP -condition, requires the existence of a linear Lyapunov function for the fluid levels of higher priority classes. Using the sufficient condition described in this paper, we can establish the existence of the 1
diffusion approximation for reentrant lines under a last-buffer-first-served (LBFS) discipline, and obtain an almost necessary and sufficient condition (except for some boundary cases) for a three-station network introduced by Dai, et al. (1999). For this latter network, we observe that the diffusion approximation may not exist, even if the “proposed” limiting SRBM exists (i.e., the corresponding reflection matrix is completely-S). At the time of this paper’s preparation, it was brought to our attention that Bramson (1998) also establishes a sufficient condition for the general heavy traffic limit theorem for the priority queueing network. The sufficient condition replaces our SHP -condition by using a uniform asymptotic stability condition; specifically, there exists a real function H(t) ≥ 0 with H(t) → 0 as t → ∞, such that, ¯ − Q(∞)|| ¯ ||Q(t) ≤ H(t) ¯ with a unit initial fluid level. We shall show that these two for all fluid level processes Q sufficient conditions are equivalent. On the other hand, the SHP -condition in our sufficient condition could be verified via a Lyapunov function approach, which leads to more explicit conditions with the verification of the feasibility of a set of linear inequalities. (We also learned that Zhang (1999) provided a relaxation of the uniform asymptotic stability condition with the ¯ → Q(∞) ¯ ¯ condition that Q(t) as t → ∞ for all fluid level process Q.) The diffusion approximation for queueing networks has long been an important area of research in queueing theory. It is the functional central limit theorem for queueing networks; and in queueing networks, it plays the same important role as the classical central limit theorem plays in the study of a sequence of i.i.d. random variables. It provides an important tool in approximating the performance measures of queueing networks. There has been a large volume of literature on the diffusion approximation. Readers are referred to Whitt (1974), Lemoine (1978), Glynn (1990) and Chen and Mandelbaum (1994) for surveys on earlier work, which is mostly on single class queueing networks. Almost all of the earlier work rely on the oblique reflection mapping (or its special case, the one-dimensional reflection mapping), introduced by Harrison and Reiman (1981). Some of multiclass queueing networks have been studied under this approach as well, including Peterson (1991) on a feedforward multiclass network, Reiman (1988) and Dai and Kurtz (1995) on a multiclass queue, and Chen and Zhang (1996) on a reentrant line queueing network with a first-buffer-first-served priority discipline. However, for general structured multiclass queueing networks under either a FIFO or a priority service discipline, the above approach may not work, since the corresponding reflection mapping may not be well-defined (or the corresponding dynamic complementarity problem may not have a unique solution); see Mandelbaum (1989) and Bernard and El Kharroubi (1991). Moreover, it is now well-known that the diffusion approximation may not exist in some cases; Dai and Wang (1993) and Dai and Nguyen (1994) provided a counterexample for a network under a FIFO service discipline, and Chen and Zhang (1998) provided a counterexample for a network under a priority discipline. Therefore, the challenge is to answer the question, “for which classes of multiclass queueing networks does the diffusion approximation exist?” In the past few years, some research has been done to address this question. Chen and Zhang (2000a) provided a sufficient condition for networks under the FIFO service discipline, and Bramson (1998) and Williams (1998b) established the diffusion approximation for networks under the head-of-the-line processor sharing service discipline and networks of Kelly type under the FIFO service discipline. As previously mentioned, the latter work also established a sufficient condition that relates the existence of the diffusion approximation to the uniformly 2
asymptotical stability. Whereas Chen and Zhang (1998b) established some more explicit sufficient conditions. In a contemporary independent work, Bramson and Dai (1999) establish the diffusion approximation for LBFS reentrant lines. Readers are also referred to Harrison (1988) and Harrison and Nguyen (1990;1993) for the earlier effort, and to Williams (1996) for the more recent survey on the diffusion approximation of multiclass queueing networks. The proof of the diffusion limit theorem for a multiclass queueing network usually takes the two steps: first to establish the C-tightness of the scaled queueing processes, and then to invoke the weak uniqueness for the limiting processes. The foundation of this approach is the work of Reiman and Williams (1988) and Taylor and Williams (1993) that provides a necessary and sufficient condition for the existence and the uniqueness of a class of semimartingale reflecting Brownian motions (SRBMs), which arises as the diffusion approximation limit (if the limit exists). A proof for the main sufficient condition that follows this approach can be found in Ye (1999). Instead, we shall provide an alternative proof by showing that our sufficient condition is equivalent to that given by Bramson (1998) and Williams (1998b). This paper is organized as follows. In the next section, we describe a multiclass queueing network and its corresponding fluid network under a priority service discipline. In section 3, we present the main sufficient condition theorem, and prove this theorem by relating it to the above-mentioned sufficient condition given by Bramson (1998) and Williams (1998). Then we shall present a number of corollaries that provide more explicit sufficient conditions. In addition, we shall present a sufficient condition for the weak stability of the multiclass priority queueing network, which implies the existence of a fluid limit (the functional strong law-of-large-numbers). In section 4, we establish the existence of the diffusion approximation for reentrant line queueing networks under the LBFS service discipline, and we study the diffusion approximation for a three-station network. We conclude our discussions in section 5. To close this section, we introduce some notation and convention that are used throughout this paper. The J-dimensional Euclidean space is denoted by RJ , its nonnegative orthant by RJ+ . Let R = R1 and R+ = R1+ . Vectors are understood to be column vectors. The transpose of a vector or a matrix is obtained by adding to it a prime. When e represents a vector, it is the vector of ones with its dimension appropriate from the context. Let x = (x1 , ..., xJ )0 ∈ RJ and a ⊆ {1, ..., J}. The scalar |a| denotes the cardinality of a and the vector xa is the restriction of x to its coordinates with indices in a. Similarly, the matrix Pab is the submatrix of a matrix P , obtained by choosing the elements with row-indices in a and column-indices in b; Paa will be abbreviated to Pa . Unless otherwise stated, vector operations and relatives are interpreted componentwise; for example, for x = (x1 , ..., xJ )0 and y = (y1 , ..., yJ )0 ∈ RJ , x > y means that each coordinate of x is strictly greater than the corresponding y coordinate. We assume the P norm in RJ ||x|| = Jj=1 |xj | for any vector x ∈ RJ . For a vector x = (xj ) ∈ RJ , diag(x) denotes a J × J dimensional diagonal matrix whose jth diagonal component is xj , j = 1, ..., J. A J × J matrix R is said to be an S-matrix, if there exists an x ≥ 0 such that Rx > 0; and it is said to be completely-S if all of its principal submatrices are S-matrices. The composition {x(y(t)), t ≥ 0} of x : R+ → RJ with y : R+ → RJ+ is the function from R+ to RJ whose jth coordinate is the real-valued function {xj (yj (t)), t ≥ 0} (j = 1, ..., J). In particular, the jth coordinate of {x(λt), t ≥ 0} with λ = (λ1 , ..., λJ )0 ∈ RJ+ is {xj (λj t), t ≥ 0}. To present our convergence result, we introduce the path space DJ [0, ∞) (the space of all functions f : [0, ∞) → RJ , which are right-continuous and have finite left limits on (0, ∞)). The path space is endowed with the Skorohod topology (see Section 3.5 of Ethier and Kurtz 3
(1986) or Pollard (1984)). For a sequence {Xn } of DJ [0, ∞)-valued stochastic processes and X ∈ DJ [0, ∞), we write Xn ⇒ X if Xn converges to X in distribution. For any x ∈ DJ [0, ∞), the uniform norm of x on the interval [s, t] is defined by ||x||[s,t] = sup ||x(u)||, s≤u≤t
with ||x||[0,t] abbreviated to ||x||t . A sequence {xn } of functions in DJ [0, ∞) is said to converge uniformly on compact set (u.o.c.) to x ∈ DJ [0, ∞), if for each t ≥ 0, ||xn − x||t → 0. This is denoted by xn → x, u.o.c. as n → ∞. We denote the probability measure of a probability space by P. We denote by σ(X) the σ-field generated by the random variables X, and by σ(B) the σ-field generated by B, a set of subsets of the sample paths. In the next subsection, σ(·) will also be used to represent a mapping from classes to stations, where its domain is the class index, integers 1, ..., K. We hope no confusion will arise from the use of σ in these two completely different contexts .
2
Queueing Network and Its Corresponding Fluid Network Model
In this section, we describe in detail the queueing network under priority service discipline and its corresponding fluid network. A rigorous description of the latter is also necessary for the formulation of the main sufficient condition theorem in the next section.
2.1
Multiclass Priority Queueing Network
The queueing network under consideration consists of J (service) stations; each station has a single server and an infinite waiting room. There are K classes of jobs; each class is served (by the server) at a specific station, while each station may serve more than one class of jobs. Let J = {1, ..., J} be the set of station indices, and let K be the set of class indices. Let σ denote a mapping from K to J , with σ(k) = j indicating that class k jobs are served at station j for each k ∈ K. Let C(j) = {k ∈ K : σ(k) = j} be the set of classes that are served at station j, j ∈ J , and let C = (cjk ) be a J × K matrix whose (j, k)th component cjk = 1 if j = σ(k) and cjk = 0 otherwise. Within a class, jobs are served in the order of arrival; and among classes, jobs are served under a preemptive resume priority service discipline. The priority discipline is described by a one-to-one mapping π from {1, ..., K} onto itself. Specifically, a class k job has priority over a class ` job if π(k) < π(`). (The mapping π can be equivalently described as a permutation (π(1), ..., π(K)) of {1, ..., K}.) The primitive data of the queueing network are a K-dimensional exogenous arrival process E = {E(t), t ≥ 0}, a K-dimensional service process S = {S(t), t ≥ 0}, and K K-dimensional routing sequences φk = {φk (n), n ≥ 0}, k = 1, ..., K. The quantity Ek (t) indicates the number of class k jobs that arrived exogenously during the time interval [0, t], and Sk (t) indicates the number of service completions for class k jobs after server σ(k) serves class k for a total of t units of time, k = 1, ..., K. We assume that Ek (t) and S` (t) (k, ` = 1, ..., K) are mutually independent renewal processes. The arrival rates of the renewal processes Ek (t) and Sk (t) are αk ≥ 0 and µk > 0 (k ∈ K) respectively. The mean interarrival time of S` (t) is denoted by mk (= 1/µk ). 4
We further assume that the interarrival times of renewal processes Ek (t) and S` (t) have finite variances a2k and b2k respectively. For convenience, let V = (Vk ), where Vk (n) indicates the sum of the service times for the first n jobs of class k, i.e., Vk (n) = inf{t : Sk (t) = n}. The quantity φk (n) = e` (the `th unit vector in RK with its `th component being one and other components being zeroes) indicates that the nth job of class k turns into a class ` job after service completion, ` ∈ K; and φk (n) = 0 indicates the nth job of class k leaves the network after service completion, k ∈ K and n ≥ 1. We assume the sequences φk , k = 1, ..., K, are mutually independent i.i.d. sequences, and they are also independent of the exogenous arrival process E(t) and the service process S(t). Let pk` be the probability that φk (n) = e` , k, ` ∈ K. We call the K × K matrix P = (pk` ) the routing matrix, and assume it has a spectral radius less than one, or equivalently, P n → 0 as n → ∞. That is, the network is an open network. Let k
Φ (0) = 0
and
k
Φ (n) =
n X
φk (i), n ≥ 1,
k ∈ K.
i=1
We assume that there are no jobs initially in the network. Before describing the performance measures and the dynamics of the queueing network, we introduce some useful notation, namely, Hk , Hk+ , L, H, h(k), B, eH , D and M . Let Hk = {` : ` ∈ C(σ(k)), π(`) ≤ π(k)} be the set of indices for all classes that are processed at the same station as class k and have a priority no less than that of class k. Let Hk+ = Hk \ {k}. We denote by L the set of all classes that have the lowest priorities at their respective stations; in other words, for each j = 1, ..., J, L ∩ C(j) 6= ∅ and for each k ∈ L, π(k) ≥ π(`) for all ` ∈ C(σ(k)), and we denote by H := K \ L. Let ( arg max{π(`) : ` ∈ Hk+ } if Hk+ 6= ∅, h(k) = 0 otherwise; in words, if k is not the highest priority class at station σ(k), then h(k) is the index for the class which has the next higher priority than class k at station σ(k), otherwise, h(k) = 0. Let B = (b`k ) be a K × K matrix with (
b`k =
1 if k = h(`), 0 otherwise,
+ H 0 H `, k = 1, 2, ..., K, and eH = (eH 1 , ..., eK ) be a K-dimensional vector with ek = 1 if Hk = ∅ and eH k = 0 otherwise. The matrix D = diag(µ) is a K-dimensional diagonal matrix whose kth element is µk , and M = D−1 = diag(m). Now we describe the performance measures and the dynamics of the queueing network under a given priority service discipline π. The performance measures of primary interest are a Kdimensional queue length process Q = (Qk ) with Qk = {Qk (t) : t ≥ 0}, and a J-dimensional workload process W = (Wj ) with Wj = {Wj (t) : t ≥ 0}, where Qk (t) indicates the number of class k jobs in the network at time t, and Wj (t) represents the amount of time that station j
5
has to work to empty out every job at station j at time t provided that no more external and internal arrivals to station j are allowed. Let Tk (t) be the total amount of time that station σ(k) has served class k jobs during (0, t]. We call T = {T (t), t ≥ 0} with T (t) = (Tk (t)) an allocation process. Note that Sk (Tk (t)) denotes the total number of class k job service completions by time t. Therefore, we have the following balance-equation (recalling our assumption that Q(0) = 0), Qk (t) = Ek (t) +
K X
Φ`k (S` (T` (t))) − Sk (Tk (t)).
(1)
`=1
The work-conserving (or non-idling) condition and the preemptive priority service discipline imply that for each k ∈ K, t
Z
Tk (t) =
0
1{Qk (s)>0 and
Q + (s)=0} ds, Hk
t ≥ 0,
(2)
where QH + (s) = 0 is assumed to hold by default whenever Hk+ = ∅. Denote the k-dimensional k total arrival process A = {A(t), t ≥ 0} as Ak (t) = Ek (t) +
K X
Φ`k (S` (T` (t))),
t ≥ 0 and k ∈ K.
`=1
Given the allocation process T and the total arrival process A, the workload process W = {W (t), t ≥ 0} can be formally defined as W (t) = CV (A(t)) − CT (t),
t ≥ 0,
(3)
where the jth coordinate reads like X
Wj (t) =
[Vk (Ak (t)) − Tk (t)].
k∈C(j)
Next, we derive some convenient alternative relations for the queue length processes. First, we rewrite (1) by centering, Q(t) = ξ(t) + αt − (I − P 0 )DT (t),
(4)
where ξ(t) = (ξk (t)), ξk (t) = [Ek (t) − αk t] − [Sk (Tk (t)) − µk Tk (t)] +
K n X
o
[Φ`k (S` (T` (t))) − p`k S` (T` (t))] + p`k [S` (T` (t)) − µ` T` (t)] .
(5)
`=1
Let Yk (t) = t −
X
T` (t),
k ∈ K,
(6)
`∈Hk
which is the cumulative idle time of the server at station σ(k) during the time interval [0, t] after serving jobs of the classes whose priorities are no less than class k. In other words, Yk (t) 6
indicates the cumulative amount of time during [0, t] that is available for station σ(k) to serve classes whose priorities are lower than class k. In particular, for k ∈ L, Yk (t) denotes the cumulative idle time of station σ(k) during the time interval [0, t]. Then the relation (6) has the vector form, T (t) = −(I − B)Y (t) + eH t. Substituting the above into (4), we obtain an alternative expression for the flow-balance relation, Q(t) = ξ(t) + θt + RY (t) ≥ 0,
(7)
where θ = α − (I − P 0 )µH ,
(8)
0
R = (I − P )D(I − B),
(9)
and µH = DeH . Next, it follows from (2) that t
Z
Yk (t) =
0
1{QH
k
(s)=0} ds,
t ≥ 0 and k ∈ K.
(10)
This clearly implies that Z 0
∞
X
k ∈ K,
Qi (t) dYk (t) = 0,
(11)
i∈Hk
which in turn implies ∞
Z 0
Qk (t)dYk (t) = 0,
k ∈ K.
(12)
It follows from (10) that for all t ≥ s ≥ 0, Yk (0) = 0
and 0 ≤ Yk (t) − Yk (s) ≤ Yh(k) (t) − Yh(k) (s)
for all k ∈ K,
where Y0 (t) ≡ t. In vector form, they are the same as Y (t) − Y (s) ≥ 0
and
Y (0) = 0, H
(I − B)[Y (t) − Y (s)] ≤ e (t − s),
(13) (14)
for all t ≥ s ≥ 0. We note that since Q ≥ 0, under (13)-(14), condition (12) is equivalent to the seemingly stronger condition (11). Let λ = (I − P 0 )−1 α, β = M λ and ρ = Cβ = CM λ. Call λ a nominal total arrival rate (vector), βk (the kth component of β) a traffic intensity for class k, k ∈ K, and ρj (the jth component of ρ) a traffic intensity for station j, j ∈ J . Usually, the vector ρ = (ρj ) is simply called the traffic intensity of the queueing network. Actually, λ is the solution to the following traffic equation, λ = α + P 0 λ, which indicates that the nominal total arrival rate vector λ includes both external arrivals and internal transitions. We say the queueing network is under a heavy traffic condition if ρ = e. 7
2.2
Multiclass Priority Fluid Network
We describe a fluid network that corresponds to the queueing network described in the previous section. One obtains the former by replacing the discrete jobs in the latter with continuous fluids. Specifically, the fluid network consists of J stations (buffers) indexed by j ∈ J := {1, ..., J}, processing (serving) K fluid (job) classes indexed by k ∈ K := {1, ..., K}. A fluid class is processed exclusively at one station, but one station may process more than one fluid classes. As in the queueing network, σ(·) denotes a many-to-one mapping from K to J , with σ(k) indicating the station at which a class k fluid is processed. A class k fluid may flow exogenously into the network at rate αk (≥ 0), then it is processed at station σ(k), and after being processed, P a fraction pk` of fluid turns into a class ` fluid, ` ∈ K, and the remaining fraction, 1 − K `=1 pk` flows out of the network. When station σ(k) devotes its full capacity to processing class k fluid (assuming that it is available to be processed), it generates an outflow of class k fluid at rate µk (> 0), k ∈ K. Let α = (αk ) and call it the exogenous inflow (arrival) rate (vector), let µ = (µk ) and call it the processing rate (vector). We call K × K substochastic matrix P = (pk` ) the flow transition matrix. Corresponding to the open queueing network described in the last subsection, we consider an open fluid network. That is, we also assume that matrix P has a spectral radius less than one. Among classes, fluid follows a priority service discipline, which is again described by a one-to-one mapping π from {1, ..., K} onto itself. Specifically, a class k has priority over a class ` if π(k) < π(`). We adopt the following notation from the queueing network subsection, C(j), C, Hk , Hk+ , h(k), L, H, B, eH , θ, R, λ, β, ρ, M , and D. We say the fluid network is under a heavy traffic condition if ρ = e. To describe the dynamics of the fluid network, we introduce the K-dimensional fluid level ¯ = {Q(t), ¯ ¯ k (t) denotes the fluid level of class k at time process Q t ≥ 0}, whose kth component Q t; and the K-dimensional unused capacity process Y¯ = {Y¯ (t), t ≥ 0}, whose kth component Y¯k (t) denotes the (cumulative) unused capacity of station σ(k) during the time interval [0, t] after serving all classes at station σ(k) which have a priority no less than class k (including class k). Sometimes, we also use a performance measure equivalent to the unused capacity, the K-dimensional time allocation process T¯ = {T¯(t), t ≥ 0}, whose kth component T¯k (t) denotes the total amount of time that station σ(k) has devoted to processing class k fluid during the time interval [0, t]. These two processes are related in the following way: Y¯k (t) = t −
T¯` (t),
X `∈Hk
and can be rewritten in the vector form, T¯(t) = −(I − B)Y¯ (t) + eH t,
(15)
With these performance measures, the dynamics of the fluid network is given as the following system of equations. ¯ = Q(0) ¯ Q(t) + θt + RY¯ (t) ≥ 0, Z 0
∞
¯ k (t)dY¯k (t) = 0, Q
Y¯ (0) = 0
and
for t ≥ 0,
for k ∈ K, DY¯ (s, t) ∈ Ψ 8
(16) (17)
for t > s ≥ 0,
(18)
where the operator Df is defined as Df (s, t) =
f (t) − f (s) t−s
for any t > s and any K-dimensional function f ; and the set Ψ is defined as Ψ = {u ∈ RK : u ≥ 0 and (I − B)u ≤ eH }.
(19)
The equations (16) and (17) can also be written in terms of the time allocation process T¯, as ¯ = Q(0) ¯ Q(t) + αt − (I − P 0 )DT¯(t) ≥ 0, for t ≥ 0, X ¯ k (t) > 0, T¯˙ k (t) = 1 if Q for t ≥ 0, k ∈ K.
(20) (21)
`∈Hk
In the fluid network (K, J , θ, R, Ψ, σ, π), we shall call θ drift vector, R reflection matrix, Ψ unused ¯ ¯ Y¯ ) (or (Q, ¯ T¯) ) is said to be a fluid solution of capacity set, and Q(0) the initial state. A pair (Q, ¯ a the fluid network (K, J , θ, R, Ψ, σ, π), if it satisfies (16)-(18). For convenience, we also call Q ¯ Y¯ ) is a fluid solution. A well-known property fluid solution if there is a Y¯ such that the pair (Q, ¯ Y¯ , and T¯ are Lipschitz continuous, and we will use later in this paper is that the processes Q, hence are differentiable almost everywhere on [0, ∞). We call a time t ∈ [0, ∞) a regular point if all derivatives of such processes exist at t. Finally, we formulate a key condition used in this paper, which is the stability of higher priority classes in the fluid network. SHP -condition. There exists a time τ > 0 such that, ¯ H (τ + ·) ≡ 0 Q ¯ satisfying (16)-(18) with the initial condition ||Q(0)|| ¯ for any fluid solution Q ≤ 1.
3
Sufficient Conditions
We first present the main result, namely the sufficient condition for the diffusion approximation. Then we prove the main theorem by showing that a key condition in the main result, namely, the SHP -condition, is equivalent to the asymptotical stability condition as described in the Introduction. Next, we present some more explicit sufficient conditions for the SHP -condition, by constructing some Lyapunov functions. Finally, we discuss the weak stability, which implies the existence of the fluid approximation for the queueing network.
3.1
The Main Result
As a standard procedure, we consider a sequence of queueing networks as described in section 2.1, indexed by n = 1, 2, · · ·. Let αn and µn be the exogenous arrival rate and the service rate respectively for the nth network. Then E n (t), with Ekn (t) = Ek (αkn t/αk ), is the exogenous arrival process, and S n (t), with Skn (t) = Sk (µnk t/µk ), is the service process associated with the nth network. For ease of exposition, we assume that the routing process does not vary with n. We assume that for each network in the sequence, the initial queue length is zero. We append 9
with a superscript n to all of the other processes and parameters that are associated with the nth network and that may vary with n. We assume that as n → ∞, √ √ n(αn − α) → ca and n(µn − µ) → cs , (22) where ca and cs are two K-dimensional constant vectors. Vectors α and µ are interpreted as the exogenous arrival rate and the service rate for the limiting network. We call λ := (I −P 0 )−1 α and ρ = CM λ, respectively, the nominal total arrival rate and the traffic intensity for the limiting network, where M = diag(m) and m = (mk ) with mk = 1/µk , k ∈ K. Let β = M λ and call its kth component βk the traffic intensity for class k, k ∈ K. Our main theorem focuses on the weak convergence of the following scaled processes: ˆ n (t) = W ˆ n (t) = Q Yˆ n (t) =
1 √ W n (nt), n 1 n √ Q (nt), n 1 n √ Y (nt), n
under the heavy traffic condition, ρ ≡ Cβ ≡ CM λ ≡ CM (I − P 0 )−1 α = e.
(23)
We next define some notation that is used in formulating the main theorem below. The following are K + 2 independent driftless K-dimensional Brownian motions: ˆ = BM (0, ΓE ) with (ΓE )i` = δi` α` a2` , E Sˆ = BM (0, ΓS ) with (ΓS )i` = δi` µ` b2` , ˆ k = BM (0, ΓkΦ ) with (ΓkΦ )i` = pki (δi` − pk` ), Φ
(24) (25) k ∈ K,
(26)
where δi` = 1 if i = `, and δi` = 0 otherwise. Let ˆ = E(t) ˆ + ξ(t)
K X
ˆ k (λk t) − (I − P 0 )S(βt); ˆ Φ
(27)
k=1
it is a K-dimensional driftless Brownian motion with covariance matrix ˆ = ΓE + Γ
K X
λk ΓkΦ + (I − P 0 )diag(β)ΓS (I − P ).
k=1
Assume that
−1 RH
ˆ = {X(t), ˆ exists. Then X t ≥ 0}, defined by −1 ˆ ˆ X(t) := ξˆL (t) − RLH RH ξH (t),
(28)
is a J-dimensional driftless Brownian motion with covariance matrix, −1 ˆ 0 −1 0 0 −1 0 ˆ L − RLH R−1 Γ ˆ ˆ Γ∗ = Γ H HL − ΓLH (RH ) RLH + RLH RH ΓH (RH ) RLH .
(29)
Let c = ca − (I − P 0 )cs,H + (I − P 0 )diag(cs )(eH − β), where cs,H = diag(cs )eH . 10
(30)
Theorem 3.1 Suppose that the convergence (22) and the heavy traffic condition (23) hold. −1 Suppose that RH exists, the covariance matrix Γ∗ , defined in (29), is non-degenerate, and ˜ L , defined by matrix R ˜ L = RL − RLH R−1 RHL R H is completely-S. If the fluid network (K, J , θ, R, σ, π) satisfies the SHP -condition, then the weak convergence ˆ n, W ˆ n , YˆLn ) ⇒ (Q, ˆ W ˆ , YˆL ), in [0, ∞), (Q
(31)
holds as n → ∞ where for t ≥ 0, ˆ H (t) = 0, Q ˆ L (t) = X(t) ˆ + ηt + R ˜ L YˆL (t) ≥ 0, Q η = cL −
−1 RLH RH cH ,
0
∞
(33) (34)
Yˆ` (·) is continuous and nondecreasing with Yˆ` (0) = 0, for all ` ∈ L Z
(32)
ˆ k (t)dYˆk (t) = 0, for all k ∈ L, Q
ˆ (t) = CM Q(t). ˆ W
(35) (36) (37)
ˆ is a martingale with respect to the filtration generated by (X, ˆ YˆL ). Moreover, the process X ˆ L = {Q ˆ L (t), t ≥ 0} is a semimartingale By Taylor and Williams (1993), the limiting process Q ˜ L ). Readers are referred to reflecting Brownian motion (SRBM) associated with (RJ+ , η, Γ∗ , R Taylor and Williams (1993) for a formal definition of the SRBM. Roughly speaking, the SRBM ˆ L starts at the origin, evolves in the interior of RJ+ like a Brownian motion with drift term Q η and covariance matrix Γ∗ , and is confined to the orthant by instantaneous reflection at the J : x = 0} is given by boundary, where the direction of reflection on the ith face {x = (xj ) ∈ R+ j ˜ the jth column of the reflection matrix RL .
3.2
Asymptotical Stability
Formally, we state the asymptotical stability condition as follows (which is a slight modification of Assumption 3.1 in Bramson (1998)). AS-condition: There is a real function H(t) ≥ 0 with H(t) → 0 as t → ∞, such that, for any ¯ Y¯ ) satisfying (16)-(18) with ||Q(0)|| ¯ (Q, ≤ 1, ¯ − Q(∞)|| ¯ ||Q(t) ≤ H(t) ¯ ¯ ¯ for all t ≥ 0 and some Q(∞) ∈ RK + with QH (∞) = 0. In addition, if QH (0) = 0, then ¯ ≡ Q(0). ¯ Q(t) It follows from Bramson (1998) and Williams (1998b) that Theorem 3.1 holds if the SHP condition is replaced by the AS-condition. The following proposition establishes the equivalence between the SHP -condition and the AS-condition. Hence, this provides a proof for Theorem 3.1.
11
˜ L = RL − RLH R−1 RHL is completely-S, the ASProposition 3.2 Under the condition that R H condition and the SHP -condition are equivalent. Before Proving this proposition, we state an oscillation inequality from Bernard and El Kharroubi (1991). Lemma 3.3 Suppose that u(·), v(·), w(·) ∈ C0J [0, ∞) (the space of all continuous functions f : [0, ∞) → RJ ) satisfy ˜ (i) w(t) Z = u(t) + RL v(t) ≥ 0, for t ≥ 0; ∞
(ii) 0
wk (t)dvk (t) = 0, for k = 1, ..., J;
(iii) u(0) ≥ 0, v(0) = 0 and Dv(s, t) ∈ Ψ for t > s ≥ 0. ˜ L ≡ RL − RLH R−1 RHL is completely-S), Then (from the assumption that matrix R H Osc(w(·), [s, t]) ≤ d1 (Osc(u(·), [s, t])), 0 ≤ s < t < ∞,
(38)
˜ L , and for any (vector) function f , where d1 is a constant depending only on the matrix R Osc(f (·), [s, t]) := sup{||f (t2 ) − f (t1 )|| : s ≤ t1 < t2 ≤ t}. Proof (of Proposition 3.2). We first show that the SHP -condition implies the AS-condition. ¯ ¯ Suppose (Q, Y ) is a fluid solution of the fluid network (K, J , θ, R, Ψ, σ, π) satisfying (16)-(18) ¯ with ||Q(0)|| ≤ 1. According to the SHP -condition, ¯ H (τ + ·) = 0. Q
(39)
From (16)-(18), we obtain ¯ L (t) = Q ¯ L (0) − RLH R−1 Q ¯ H (0) + RLH R−1 Q ¯ H (t) + R ˜ L Y¯L (t), Q H H Z 0
∞
¯ k (t)dY¯k (t) = 0, Q
Y¯ (0) = 0
and
for k ∈ L, DY¯ (s, t) ∈ Ψ
(40) (41)
for t > s ≥ 0,
(42)
where we use −1 θ˜L := θL − RLH RH θH = 0,
(43)
which follows from Lemma 2.2 in Chen and Zhang (2000b) due to the fact that the traffic intensity ρ = e. Combined with (39), equation (40) implies ¯ L (τ + t) = Q ¯ L (τ ) + R ˜ L (Y¯ (τ + t) − Y¯ (τ )). Q Then applying Lemma 3.3, we have ¯ L (τ + ·), [s, t]) = 0, 0 ≤ s < t < ∞, Osc(Q i.e., ¯ L (τ + ·) = Q ¯ L (τ ). Q
12
¯ ¯ ), and let Now, we let Q(∞) = Q(τ (
H(t) =
M (τ − t), 0 ≤ t ≤ τ, 0, t > 0,
¯ Then it can be seen that the where M > 0 is a Lipschitz constant for the fluid level process Q. first half of AS-condition holds. ¯ H (0) = 0. We would like to show that Q ¯ H (t) = 0 for all t ≥ 0. Otherwise, Now, suppose Q τ τ ¯ ¯ let τ be as defined in the SHP -condition and define (Q , Y ) as (
¯ τ (t) = Q
¯ Q(0) t ∈ [0, τ ] ¯ − τ ) t ∈ [τ, ∞) Q(t
and (
Y¯kτ (t) =
(1 − `∈Hk β` )t t ∈ [0, τ ] τ ¯ ¯ Yk (τ ) + Yk (t − τ ) t ∈ [τ, ∞) P
for k ∈ K.
¯ τ , Y¯ τ ) satisfies (16)-(18) with ||Q ¯ τ (0)|| ≤ 1. According Then, it can be verified directly that (Q ¯ τ (τ + ·) = 0, and hence Q ¯ H (·) ≡ 0. Using equation (40) to the SHP -condition, we know that Q H ¯ ¯ and Lemma 3.3 again, we have Q(t) ≡ Q(0). Next we show that the AS-condition implies the SHP -condition. Denote a constant d2 as d2 = d1 ×
X
|rij |,
−1 where rij is the (i, j)-th element of the matrix RLH RH , and d1 is the constant defined in Lemma 3.3. Define a real number as 1 , = 2(K − J + d2 J)
and let t be a number such that H(t + t) < for all t ≥ 0. Then, by the AS-condition, we have ¯ H (t + t)|| ≤ ||Q(t ¯ + t) − Q(∞)|| ¯ ||Q ≤ H(t + t) < , for all t ≥ 0, and this inequality, combined with (40)-(42) and Lemma 3.3, implies ¯ L (·), [s, t]) ≤ d1 · Ocs(RLH R−1 Q ¯ H (·), [s, t]) < d2 Ocs(Q H for s, t ≥ t . ¯ 1 , T¯1 ) from (Q, ¯ T¯) as the following: Now, let us construct a fluid solution (Q ¯ k (t + t), Q
k ∈ H, ¯ k (t + t), ¯ k (t ) ≤ d2 , Q k ∈ L, Q Q ¯ k (t + t) − Q ¯ k (t ) + d2 , k ∈ L, Q ¯ k (t ) > d2 , Y¯ 1 (t) = Y¯ (t + t) − Y¯ (t ) ¯ 1k (t) = Q
13
¯ 1 , T¯1 ) satisfies (16)-(18), and It can be checked directly that (Q ¯ 1 (0)|| = ||Q ¯ 1 (0)|| + ||Q ¯ 1 (0)|| ≤ (K − J) + Jd2 = 1 . ||Q H L 2 If we define ¯ 2 (t) = 2Q ¯ 1 ( t ), Q 2 t 2 1 Y¯ (t) = 2Y¯ ( ), 2 ¯ 2 , T¯2 ) is a fluid solution of fluid network (16)-(18) and then it can be verified directly that (Q ¯ 2 (0)|| ≤ 1; and hence, according to the AS-condition, ||Q ¯ 2H (t + t) < Q holds for all t ≥ 0. This directly leads to ¯ H (t + t + t) < Q 2 2 for all t ≥ 0. By inductively using the above technique, we have n X
¯ H( Q
2−i t + t)
0 such that for any Q 0 0 ¯ ¯ (16)-(18), the function h QH (·) is strictly decreasing at t wherever h QH (t) > 0. It is clear that the L2 -condition in fact constructs a linear Lyapunov function for the stability of higher priority classes. Proposition 3.4 The L2 -condition implies the SHP -condition. Proof. Suppose the L2 -condition is true. According to remarks after Theorem 3.1 in Chen ¯ H (t) is compact. Combined with the and Zhang (2000b), the set of all possible derivative of Q 2 L -condition, we know that there exists > 0 such that ¯˙ H (t) < − h0 Q 14
¯ H (t) 6= 0 and t is a regular point (i.e., the point at which derivatives of all functions when Q ¯ H (0)|| ≤ 1, then h0 Q ¯ H (0) ≤ ||h||, and hence h0 Q ¯ H (t) = 0 must hold for all involved exist). If ||Q t > ||h||/. This establishes the SHP -condition. 2 Remark The converse of Proposition 3.4 is not true. Even if both the SHP -condition holds and ˜ L is completely-S, the L2 -condition may not hold. We construct a counterexample the matrix R at the end of subsection 4.2. Therefore, the sufficient condition given by Theorem 3.1 strictly improves the sufficient condition given by Chen and Zhang (2000b). Next, similar to the above explicit L2 -condition, we transplant the piecewise linear Lyapunov function approach for the stability of priority fluid network in Chen and Ye (1999) to construct another explicit condition for the existence of the diffusion approximation for the multiclass queueing network with priority service discipline. To this end, we introduce some notation and terminology. A partition (a, b) of K is called a (priority π) hierarchical partition, which shall be written as (a, b) ∈ PK , if ` ∈ a implies k ∈ a, provided that σ(k) = σ(`) and π(k) < π(`); in words, if ` ∈ a, and k is a class with higher priority in the same station as `, then k ∈ a as well. The head class set H(B) of B ⊆ K is defined to be the set {k ∈ K : if ` ∈ B and σ(`) = σ(k), then π(k) < π(`)}; in words, the set of the highest priority classes in each station that serves at least one class in B. If (a, b) ∈ PK is a hierarchical partition, then the fluid state S(a, b) is defined to be the set {q ∈ RK : q ≥ 0, qa = 0 and qH(b) > 0}, and the regular flow rate set F (a, b) is defined to be {d =
0 db
!
∈ RK : db = θb + Rba ya where ya satisfies 0 ≤ ya ≤ e and θa + Ra ya = 0}.
¯ It follows from the dynamic relations (16)-(18) that for (a, b) ∈ PK , if Q(t) ∈ S(a, b) and t is a ˙ ¯ regular point, then the derivative Q(t) must be an element in F (a, b). (In fact, it is not difficult to show by construction that the set F (a, b) is exactly the set of all such derivatives.) Now we formulate the sufficient condition based on the piecewise linear Lyapunov function, and we shall refer to it as the P L2 -condition in this paper. Similar results for the stability of multiclass fluid network with priority service discipline are contained in Theorem 3.1 and Corollary 3.2 in Chen and Ye (1999). P L2 -condition: There exists an > 0 and J (K −J)-dimensional nonnegative vectors x1 , ..., xJ such that the following two conditions hold: (a) For any j ∈ {1, ..., J}, and any hierarchical partition (a, b) with b ∩ C(j) ∩ H 6= φ and F (a, b) 6= φ, sup x0j dH ≤ −. (44) d∈F (a,b)
(b) For any partition J0 ∪ J1 = J with J0 and J1 being nonempty, denote K0 = {k : σ(k) ∈ J0 } ∩ H and K1 = (K ∩ H) \ K0 , then for any j0 ∈ J0 , there exists j1 ∈ J1 such that (xj0 )K1 ≤ (xj1 )K1 .
15
(45)
Proposition 3.5 The P L2 -condition implies the SHP -condition. Proof. Without lost of generality, assume that ¯ = /2
and
PJ
j=1 xj
x ¯j = xj + (
> 0. (Otherwise, let
)e, 2M
j = 1, · · · , J,
with M > 0 being a sufficiently large constant (say, a Lipschitz constant for the fluid level ¯ Then, it is direct to check that the P L2 -condition is still satisfied with , x1 , · · · , xJ process Q). being replaced by ¯, x ¯1 , · · · , x ¯J .) Let ¯ H (t) fj (t) = x0j Q and f (t) = max fj (t). 1≤j≤J
It suffices to show that, under the assumption of the P L2 -condition, ¯ = 0, f (t) = 0 if and only if Q(t) and ˙ ¯ f (t) ≤ − whenever Q(t) 6= 0 and f is differentiable at t.
(46) (47)
Note that N i=1 xi > 0 clearly implies the condition (46). Hence, it suffices to show (47). The remaining proof draws a similar idea from the proof of Lemma 4.1 in Dai, et al. (1999). [Also see the proof of Theorem 3.1 in Chen and Ye (1999).] ¯ H (t0 ) 6= 0 (or equivalently f (t0 ) 6= 0). Let I0 be a Suppose t0 ≥ 0 is a regular point and Q subset of {1, · · · , J} such that P
fi (t0 ) = f (t0 ) for i ∈ I0
fi (t0 ) < f (t0 ) for i 6∈ I0 .
and
¯ 0 ) ∈ S(a0 , b0 ), i.e., Let (a0 , b0 ) be the hierarchical partition such that Q(t ¯ i (t0 ) = 0, and Q ¯ k (t0 ) = 0 for any k such that σ(k) = σ(i) and π(k) < π(i)}, a0 = {i : Q ¯˙ 0 ) ∈ F (a0 , b0 ). Let and b0 = K \ a0 . Then we note Q(t J1 = {j : b0 ∩ C(j) ∩ H = 6 φ}, K0 = {k : σ(k) ∈ J0 } ∩ H,
J0 = J \ J1
K1 = (K ∩ H) \ K0 .
It follows from the condition (45) that ¯ K (t0 ) max fj (t0 ) = max(xj )0K1 Q 1 j ∈J / 1
≤
j ∈J / 1 0 ¯ max(xi )K1 QK1 (t0 ) i∈J1
= max fi (t0 ). i∈J1
This combined with the definition of I0 implies that J1 ∩ I0 6= ∅. Let i0 ∈ J1 ∩ I0 ; by condition (44), we have f˙i0 (t0 ) ≤ −.
16
Let {t1m } ⊆ [0, t0 ), {t2m } ⊆ (t0 , ∞) be two sequences such that t1m → t0 and t2m → t0 when m → ∞. Then fi0 (t0 ) − fi0 (t1m ) f (t0 ) − f (t1m ) ≤ , t0 − t1m t0 − t1m f (t2m ) − f (t0 ) fi (t2 ) − fi0 (t0 ) ≥ 0 m2 , 2 tm − t0 t m − t0 since f (t) ≥ fi0 (t) and f (t0 ) = fi0 (t0 ). Letting m → ∞, we have f˙(t0 ) = f˙i0 (t0 ) ≤ −.
2
Similar to the use of the piecewise linear Lyapunov function approach for the stability of multiclass fluid network with priority service discipline in Chen and Ye (1999), we could formulate the P L2 -condition in a more general form. We omit the extension here. The P L2 -condition seems quite involved; however, it could be formulated into the problem of solving a set of linear programs. To this end, we introduce some notation. A linear combination Pp Pp K n=1 ξn zn is called a convex combination of vectors z1 , · · · , zp in R if ξn ≥ 0 and n=1 ξn = 1. K A set Z ⊆ R is called a convex set if any convex combination of finite points of set Z is still in Z. A point z ∈ RK is said to be an extreme point of the convex set Z if z ∈ Z but is not a convex combination of other points in Z. It can be checked directly by definition that the regular flow rate set F (a, b) is a bounded convex set with at most finite extreme points. Denote the set of extreme points of F (a, b) as F e (a, b). Clearly, F e (a, b) = F (a, b) if F (a, b) is an empty set or a single point set. Then from Krein-Milman Theorem (see for example Rockafellar (1970)), any element q ∈ F (a, b) is a convex combination of the points in F e (a, b). From this result, we can easily linearize the condition (a) of the P L2 -condition. Furthermore, denote all the partitions of J into two nonempty sets as J = J0n ∪ J1n , n = 1, · · · , p, with p = 2J − 2, and define K0n and K1n as in (b) of the P L2 -condition. Now, we convert the proposition to a set of LP problems: For any sequence of mappings j1n = j1n (·) : J0n → J1n , n = 1, · · · , p, we first solve the following LP problem max s.t.
j 1 ,···,j p 1
x0j q
1
≤ −j 1 ,···,j p , for any q ∈ F e (a, b); 1
(48)
1
for any hierarchical partition (a, b) with b ∩ C(j) ∩ H = 6 φ, (xj0 )K1n ≤ (xj1n (j0 ) )K1n , for any j0 ∈ J0n , n = 1, · · · , p, J X
||xi || = 1,
(49) (50)
i=1
x1 , · · · , xJ ≥ 0.
(51)
Suppose the optimal values of the above LP problems are ∗j 1 ,···,j p respectively. Then, the P L2 1 1 condition is satisfied if and only if ∗ =
max
j11 (·),···,j1p (·)
∗j 1 ,···,j p > 0. 1
1
In this LP problems formulation, the series of partitions of station index set J are considered because the condition (b) of the P L2 -condition should be satisfied for any partition of J into two nonempty sets. That each sequence of mappings {j11 (·), · · · , j1p (·)} corresponds to a LP problem 17
is incurred by the flexibility of choice of index j1 in the condition (b) of the P L2 -condition. The constraints (48) and (49) are translated from (44) and (45) respectively. At last, the constraint (50) is a restriction of the flexibility of the choice of the number and xij ’s. We further explain why the P L2 -condition holds if and only if ∗ > 0. Suppose the P L2 condition holds. Without loss of generality, suppose that (50) is also satisfied. According to the condition (a) in the P L2 -condition, it is easy to see that constraint (48) is satisfied with j 1 ,···,j p replaced by . By the condition (b) in the P L2 -condition, we can construct a sequence 1 1 of mappings {ˆj11 (·), · · · , ˆj1p (·)} (from J0n to J1n ), such that, for each partition J0n ∪ J1n and any j0 ∈ J0n , inequality (45) holds with j1 and K1 replaced by ˆj1n (j0 ) and K1n . That is, (49) is satisfied with j1n replaced by ˆj1n (j0 ). Now, , x1 , · · · , xJ given in the P L2 -condition is a feasible solution to the LP problem corresponding to the sequence of mappings {ˆj11 (·), · · · , ˆj1p (·)}. Hence, this LP problem has a positive optimal solution, and thus ∗ > 0. On the other hand, suppose that ∗ > 0. Then, we choose one of the LP problems with positive optimal solution. Suppose this LP problem corresponds to a sequence of mappings ˆj11 (·), · · · , ˆj p (·). We claim that, if we denote the optimal solution of this LP problem as x1 , · · · , xJ , 1 and , the P L2 -condition holds with this set of parameters. Actually, all the conditions in the P L2 -condition are obvious except the condition (b). Consider any nonempty partition J = J0 ∪ J1 in the condition (b). We have J0 = J0n , J1 = J1n , K0 = K0n , and K1 = K1n for some n ∈ {1, · · · , p}. For any j0 ∈ J0 , let j1 = ˆj1n (j0 ), then, according to the constrain (49) of the LP problem, we know at once that (45) is true.
3.4
Weak Stability
Here we discuss the relationship between the SHP -condition and the weak stability of the fluid network, since the concept of weak stability is closely related to the existence of diffusion approximation for the queueing network. A fluid network (K, J , θ, R, Ψ, σ, π) is said to be weakly ¯ = 0 and Y¯ (t) = −R−1 θt under the initial condition stable if (16)-(18) has a unique solution Q(t) ¯ Q(0) = 0. It is known (see Chen (1995)) that the weak stability implies that the fluid limit, 1 n Q (nt) → 0, n
u.o.c. as n → ∞,
almost surely, and this convergence is also a necessary condition for the existence of the diffusion approximation for the multiclass priority queueing network (see Chen and Zhang (2000b)). Theorem 3.6 Suppose that the traffic intensity of the fluid network, ρ ≡ CM (I − P 0 )−1 α = e, −1 ˜ L = RL − RLH R−1 RHL is completely-S. Then the fluid that RH exists, and that the matrix R H network is weakly stable if it satisfies the SHP -condition. Remark. The converse of Theorem 3.6 is not true. In Proposition 4.4, an example will be given to show this. ¯ Proof of Theorem 3.6. Suppose the initial fluid level of the fluid network is Q(0) = 0. First, we show by contradiction that ¯ H (t) = 0, Q
for all t ≥ 0.
Let ¯ H (t) 6= 0}. t0 = inf{t ≥ 0 : Q 18
(52)
¯ H (t) = 0 on [0, t0 ], we have Then, keeping in mind that Q ¯ L (t) = 0, Q
for 0 ≤ t ≤ t0 ,
¯ 0 ) = 0. by equations (40)-(42) and Lemma 3.3. Specifically, we have Q(t According to the definition of t0 , there exists a t1 > t0 such that ¯ H (t1 )|| = ||Q
(53)
¯ 2 ) 6= 0, it can be checked directly that for some > 0. However, for any t2 > t0 with Q(t ¯ ∗ (t) = Q
1 ¯ 2 + ||Q(t ¯ 2 )||t) Q(t ¯ ||Q(t2 )||
is also a fluid solution satisfying (16)-(18). Hence, according to the SHP -condition, we have ¯ ∗H (τ + ·) ≡ 0, Q which immediately leads to ¯ H (t2 + ||Q(t ¯ 2 )||τ + ·) ≡ 0. Q ¯ 2 ) 6= 0) sufficiently close to t0 , then we get t2 + ||Q(t ¯ 2 )||τ < t1 , If we choose a point t2 (with Q(t ¯ ¯ ¯ since Q(t0 ) = 0 and Q(·) is continuous. Hence, ||QH (t1 )|| = 0. But this contradicts (53) and thus proves (52). −1 Finally, according to Lemma 3.3, the completely-S of RL − RLH RH RHL implies that (40)¯ (42) has a unique solution QL (t) ≡ 0. 2
4
Examples
In this section, we apply the main theorem of the previous section to study the diffusion approximation of two queueing network examples. In the first subsection, we prove the existence of the diffusion approximation for the reentrant line network under the LBFS service discipline. In the second subsection, we characterize the condition for the existence of the diffusion approximation for a three-station queueing network which is first studied in Dai, et al (1999) and which will be referred to as DHV network. We also employ a variation of this network to provide a counterexample for Proposition 3.4.
4.1
Reentrant Line with Last Buffer First Served Discipline
We consider a reentrant line network under the LBFS discipline, i.e., π = (K, K −1, ..., 2, 1). The parameters of the reentrant line network take the following special form: the exogenous inflow rate α = (1, 0, ..., 0)0 and the flow-transfer matrix P = (pik ) with pk,k+1 = 1 for k = 1, ..., K − 1 and pik = 0 for all other i, k ∈ K, as shown by Figure 1. In this case, we have θ = (θ1 , · · · , θK )0 H H with θk = µH k−1 − µk (µ0 := 1) for k = 1, · · · , K; and R = (Rk` )K×K with R11 = µ1 , R1` = −µ1 b1` , ` = 2, · · · , K, 19
and, for k = 2, · · · , K, Rk` = 0, ` = 1, · · · , k − 2, Rk,k−1 = −µk−1 , Rkk = µk−1 bk−1,k + µk , Rk` = µk−1 bk−1,` − µk bk` , ` = k + 1, · · · , K, where (bk` ) is as defined in subsection 2.1.
-
-
-
-
-
α1 = 1 -
-
-
6
Figure 1: A Re-entrant Line Network We note that in this case, the equation (20) takes the following simpler form, ¯ k (t) = Q ¯ k (0) + µk−1 T¯k−1 (t) − µk T¯k (t), Q
k ∈ K,
(54)
where µ0 = 1 and T¯0 (t) ≡ t. ˜ L = RL − RLH R−1 RHL is completely-S (see Dai, It is known that when ρ = e, the matrix R H Yeh and Zhou (1997), Theorem 3.1). The next proposition implies that the network satisfies the SHP -condition under ρ = e. (The proposition appears slightly stronger than the SHP ¯ ¯ H (0)|| ≤ condition, since the initial condition ||Q(0)|| ≤ 1 in the SHP -condition is replaced by ||Q 1.) Proposition 4.1 Suppose that the traffic intensity, ρj ≡ k∈C(j) mk = 1 for all j ∈ J . Then, ¯ H (τ + ·) ≡ 0, for any fluid solution (Q, ¯ T¯) of the there exists a real number τ ≥ 0 such that Q ¯ reentrant line with ||QH (0)|| ≤ 1. P
The proof of Proposition 4.1 is based on the following Lemma. To state the lemma, we introduce some notation. Let 1 = k1 < k2 < ... < kJ ≤ K be the J classes in the lower priority ¯ Denote the set of class set L, and M > 0 the Lipschitz constant for the fluid level process Q. classes between class kj and kj+1 as Aj = {k ∈ K : kj < k < kj+1 }, where j = 1, ..., J, and kJ+1 := K + 1 for convenience. 20
¯ A (τj + ·) ≡ Lemma 4.2 Let j ∈ J be fixed. Then, there exists a real number τj ≥ 0 such that Q j ¯ T¯) of the reentrant line with 0, for any fluid solution (Q, ¯ A (0)|| ≤ 1 and ||Q ¯ ∪ A (t)|| ≡ 0. ||Q j i>j i
(55)
¯ A (t) ≡ 0 if A = φ.) (By default, Q Proof.
We prove the lemma in three steps.
¯ T¯) of the reentrant line satisfying Step 1. We first show that, for any given fluid solution (Q, (55), ¯ A (t) = 0} < ∞. inf{t : Q (56) j ¯ A (t)|| > 0 then It is sufficient to show that, for any regular point t > 0, if ||Q j d ¯ ||Q (t)|| < −γ dt {`:`≤kj+1 −1} for some constant number γ > 0. ¯ A (t)|| > 0. Let k ∗ = max{k : Q ¯ k (t) > Now, suppose that t > 0 is a regular point and that ||Q j ¯ 0, k ∈ Aj }, and B = {k ∈ L : k > kj , Qk (t) > 0}. Then, by equation (17), we have T¯˙ k (t) = 1
X
for all ` ∈ {k ∗ } ∪ B,
(57)
k∈H`
¯ ` (t) > 0 for such class `. For class ` ∈ ¯ ` (t) = 0, Q ¯ ` (·) is a since Q / B and k ∗ < ` ≤ K, since Q ˙ ¯ ` , it must be that Q ¯ ` (t) = 0. This, together non-negative function and t is a regular point of Q with equation (54), implies µ`−1 T¯˙ `−1 (t) = µ` T¯˙ ` (t)
for all ` ∈ / B and k ∗ < ` ≤ K.
(58)
Let `∗ = min{` : ` ∈ B ∪ {K + 1}}. Then, solving the K − `∗ + 1 equations corresponding to ` ≥ `∗ in (57) and (58) jointly, we have µ` T¯˙ ` (t) = 1
for `∗ ≤ ` ≤ K.
(59)
The equation in (57) with ` = k ∗ can be written as m`0 (µ`0 T¯˙ `0 (t)) +
X `0 ∈Hk∗ ,`0 1, and
µ` T¯˙ ` (t) ≥ 1
for ` ∈ Hk+j+1 .
However, this contradicts T¯˙ ` (t) ≤ 1.
X `∈Hkj+1
¯ T¯) of the Step 2. Next, we show that, there exists τj ≥ 0 such that, for all fluid solution (Q, reentrant line satisfying (55), we have ¯ A (t) = 0} ≤ τj . inf{t : Q (65) j ¯ A (t) = 0}. We first estimate µk T¯k (t∗ ), which is the total outflow of class Let t∗ = inf{t : Q j j j kj fluid up to time t∗ . For each k ∈ Hkj and k < kj+1 , we have ¯ {`:k t∗ such that ||Q j j ∗ 0} is open and nonempty, there exists an interval [t1 , t2 ] ⊂ [t , +∞) such that ¯ A (t1 )|| = 0; and ||Q ¯ A (t)|| > 0 for t ∈ (t1 , t2 ). ||Q j j ¯ A (t)||. Let Let be a number such that 0 < < supt1 ≤t≤t2 ||Q j ¯ A (t)|| = }. t = min{t > t1 : ||Q j Now, define the following: ¯ (t) = 1 Q(t ¯ + t), Q 1 T¯ (t) = T¯(t + t).
23
(71)
¯ , T¯ ) is also a fluid solution of the reentrant line satisfying It can be checked directly that (Q condition (55). Then, by (65), ¯ (t)|| = 0} ≤ τj , inf{t > 0 : ||Q Aj and this immediately leads to ¯ A (t)|| = 0} ≤ t + τj . t¯ = inf{t > t : ||Q j Since t → 0 when → 0, we have t¯ ∈ (t1 , t2 ) for sufficient small . But this contradicts condition (71). 2 Proof of Proposition 4.1. We use the above lemma inductively to show that, there exist fixed times tJ ≤ tJ−1 ≤ ... ≤ t1 such that ¯ A (tj + ·) ≡ 0, for j = J, · · · , 1, Q j
(72)
¯ T¯) of the reentrant line with ||Q ¯ H (0)|| ≤ 1. First, let tJ = τJ in Lemma for any fluid solution (Q, ¯ A (tJ + ·) ≡ 0, since ||Q ¯ A (0)|| ≤ ||Q ¯ H (0)|| ≤ 1. Next, suppose Q ¯ A (ti + ·) ≡ 0, for 4.2; then Q i J J 1 1 ¯ ¯ i = J, · · · , j + 1 (j ≥ 1). Consider the pair (Q , T ) defined by 1 ¯ j+1 + (1 + M tj+1 )t), Q(t 1 + M tj+1 1 T¯1 (t) = T¯(tj+1 + (1 + M tj+1 )t), 1 + M tj+1
¯ 1 (t) = Q
(73)
¯ Under the induction assumption, it can be checked where M is a Lipschitz constant of Q. 1 1 ¯ ¯ directly that (Q , T ) is a fluid solution of the reentrant line satisfying (55), since ¯ 1A (0)|| = ||Q j
1 1 ¯ A (tj+1 )|| ≤ ¯ A (0)|| + M tj+1 ) ≤ 1. ||Q (||Q j j 1 + M tj+1 1 + M tj+1
Then, by Lemma 4.2, ¯ 1 (τj + ·) ≡ 0. Q Aj
(74)
¯ A (tj + ·) ≡ 0. Let tj = tj+1 + (1 + M tj+1 )τj . We have immediately, by (73) and (74), that Q j Now, τ = t1 is what we want in this proposition. 2 In view of Theorem 3.1, summarizing the above, we have Theorem 4.3 Suppose that, for the reentrant line with LBFS discipline, the convergence (22) and the heavy traffic condition ρ = e hold. Then the weak convergence (31) holds with the limits ˆ is a martingale with respect to the defined by equations (32)-(37). Moreover, the process X ˆ ˆ filtration generated by (X, YL ).
24
Station 1
Station 2
Station 3
α1 = 1-
m1
-
m2
-
m3
-
m4
-
m5
-
m6
-
Figure 2: The three-station fluid network example
4.2
A Three-Station Priority Network
We consider the network as shown by Figure 2, which shall be referred to as DHV network since it was first studied by Dai, Hasenbein and Vande Vate (1999). The parameters α, P and C for this network take the form, α=
1 0 0 0 0 0
1 0 0 , C = 0 1 0 0 0 1
1 0 0 0 1 0 , P = 0 0 1
0 0 0 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
.
The priority mapping (permutation) is represented by π = (4, 1, 5, 2, 6, 3); specifically, classes 4, 2 and 6 have higher priorities at stations 1, 2 and 3 respectively. In this case, we have θ=
1 −µ2 µ2 −µ4 µ4 −µ6
,
R=
µ1 0 0 −µ1 0 0 −µ1 µ2 0 µ1 0 0 0 −µ2 µ3 0 0 −µ3 , 0 0 −µ3 µ4 0 µ3 0 −µ5 0 −µ4 µ5 0 0 µ5 0 0 −µ5 µ6
and, under heavy traffic condition ρ = e,
˜ L = RL − RLH R−1 RHL R H
m3 m5 −m4 m5 m4 m6 1 m5 −m6 . = m2 − m3 m1 m3 m5 + m2 m4 m6 −m2 m2 − m1 m1
Proposition 4.4 Under the heavy traffic condition that ρ = e (i.e., ρ1 = m1 + m4 = 1, ρ2 = m2 + m5 = 1 and ρ3 = m3 + m6 = 1), 25
˜ L is completely-S; (i) R (ii) the DHV fluid network satisfies the SHP -condition if and only if m2 + m4 + m6 < 2; (iii) the DHV fluid network is weakly stable if and only if m2 + m4 + m6 ≤ 2. Proof of Proposition 4.4 (i). By definition of the completely-S matrix and in view of the heavy traffic condition, it suffices to verify the following: ˜ L are all positive. 1. The diagonal elements of the matrix R ˜ {1,2} x > 0, i.e., 2. There exists a vector x ∈ R2+ and x > 0, such that R (
m3 m5 x1 − m4 m5 x2 > 0, (m2 − m3 )x1 + m5 x2 > 0.
To see this, first note the above is equivalent to the existence of positives x1 and x2 such that m5 m3 x1 > m5 m4 x2 > m4 (m3 − m2 )x1 . The latter is further equivalent to m5 m3 > m4 (m3 − m2 ), which, by some simple algebra and the heavy traffic assumption, can be written as m2 (1 − m2 ) > (m2 − m1 )(m3 − m2 ). We claim that the above inequality holds, since either m2 > m2 − m1 > 0 and 1 − m2 > m3 −m2 , or m2 > m2 −m3 and 1−m2 > m1 −m2 ≥ 0 hold. From the above process, we can also see that the vector x can be chosen as x1 = 1 and x2 ∈ (max(µ5 (m3 − m2 ), 0), µ4 m3 ). Similarly, we can show that ˜ {1,3} x > 0 holds for x1 = µ2 and x2 > µ1 ; 3. R ˜ {2,3} x > 0 holds for x1 = 1 and x2 ∈ (max(µ1 (m1 − m2 ), 0), µ6 m5 ); 4. R ˜ L x > 0 holds for x2 = 1, x1 = δ > 0 (a sufficient small number) and 5. R x3 ∈ (max(−µ4 µ6 m3 m5 δ + µ6 m5 , µ1 m2 δ + µ1 (m1 − m2 ), 0), µ6 (m2 − m3 )δ + µ6 m5 ) . 2 Proof of Proposition 4.4 (ii) and (iii).
We break down the proof into four lemmas.
Lemma 4.5 Under the heavy traffic condition, the DHV fluid network satisfies the SHP condition if m2 + m4 + m6 < 2. In Dai, et al. (1999), it is shown that the linear Lyapunov condition given by Chen and Zhang (2000b) could not provide a sharp characterization for the stability of the DHV network (under ρ < e). However, the linear Lyapunov condition given by Chen and Zhang (2000b) does yield a sharp characterization (except for a boundary case) for the diffusion approximation of the DHV network (under the heavy traffic condition ρ = e); see Theorem 4.10. In particular, we shall prove the lemma by using Theorem 3.2 in Chen and Zhang (2000b) (in view of Proposition 3.4), and for convenience, we quote it below. 26
Theorem 4.6 Suppose that there exists a (K − J)-dimensional vector h > 0 such that for any given partition a and b of K satisfying (i) If class ` ∈ a, then each class k with σ(k) = σ(`) and π(k) > π(`) is also in a, (ii) a ∩ H = 6 ∅, we have h0a∩H (θa + Rab xb )a∩H < 0, for xb ∈ Sb := {u ≥ 0 : θb + Rb u = 0 and u ≤ e} where b 6= ∅, and xb = 0 when b = ∅. (The inequality is assumed to hold by default when Sb = ∅). Then the network satisfies ¯˙ H (t) < 0 for almost all t ≥ 0 and any Q ¯ satisfying equation (16)-(18) (with the (1) h0 Q ¯ parameters specified in this example) with ||QH (t)|| > 0 , and hence (2) the L2 -condition. Proof of Lemma 4.5. Through a tedious analysis of a set of inequalities corresponding to all the possible partitions a ∪ b in Theorem 4.6, we show that there exists a vector h as described in the above theorem if and only if there exists an h satisfying the following system of inequalities: −µ2 h2 + (µ3 − µ4 )h4 < 0
(75)
−µ6 h6 + (µ1 − µ2 )h2 < 0
(76)
−µ4 h4 + (µ5 − µ6 )h6 < 0.
(77)
In order to show the existence of such an h satisfying the inequalities (75)-(77), we divide the proof into the following four cases and show the existence of such an h in each of the four cases. The four cases are based on the following representation: {m : ρ = e, m2 + m4 + m6 < 2} = {m : ρ = e, m2 ≤ m1 } ∪ {m : ρ = e, m4 ≤ m3 } ∪ {m : ρ = e, m6 ≤ m5 } ∪{m : ρ = e, m2 > m1 , m4 > m3 , m6 > m5 , m2 + m4 + m6 < 2}. Case 1. Suppose m2 ≤ m1 . Now, we select h2 = δ, h4 = M, h6 = 1, where M is a sufficiently large positive number and δ a sufficiently small positive number. Then, h satisfies the inequalities (75)-(77). Case 2. Suppose m4 ≤ m3 . Similar to case 1, we select in this case, h2 = M, h4 = 1, h6 = δ, where M is a sufficient big positive number and δ a sufficient small positive number. Case 3. Suppose m6 ≤ m5 . In this case, we select h2 = 1, h4 = δ, h6 = M, 27
where M is a sufficient big positive number and δ a sufficient small positive number. Case 4. Suppose m2 > m1 , m4 > m3 , m6 > m5 and m2 + m4 + m6 < 2. In this case, we show the existence of h directly. Under the assumption of this case, we have µ1 − µ2 > 0, µ3 − µ4 > 0 and µ5 − µ6 > 0. By eliminating h2 from inequalities (75) and (76), we can show that the existence of positive constants h2 , h4 and h6 satisfying inequalities (75)-(77) is equivalent to the existence of positive h4 and h6 satisfying (µ1 − µ2 )(µ3 − µ4 )h4 < µ2 µ6 h6 ,
and
−µ4 h4 + (µ5 − µ6 )h6 < 0. It is clear that the above is equivalent to (µ1 − µ2 )(µ3 − µ4 )(µ5 − µ6 ) < µ2 µ4 µ6 . Note that mk = 1/µk ; the above is the same as (m2 − m1 )(m4 − m3 )(m6 − m5 ) < m1 m3 m5 . But m1 m3 m5 − (m2 − m1 )(m4 − m3 )(m6 − m5 ) = (m1 + m3 − m2 ) ((m2 − m1 )m6 + m1 m5 ) >0 since m1 + m3 − m2 = 2 − m2 − m4 − m6 > 0 and m2 − m1 > 0 under the assumption in this case. 2 Lemma 4.7 Under the heavy traffic condition, the DHV fluid network is weakly stable if m2 + m4 + m6 ≤ 2. Proof. By an almost word for word repetition of the proof of Lemma 4.5, we can show that, ¯˙ H (t) = h0 Q ¯˙ {2,4,6} (t) ≤ 0 for almost all there exists a 3-dimensional vector h > 0 such that h0 Q ¯ satisfying equations (16)-(18) with parameters for the DHV fluid network. This t ≥ 0 and any Q ¯ establishes that if the initial condition is Q(0) = 0, then ¯ H (t) = 0 for all t ≥ 0. Q Then, following the same line of the proof of Theorem 3.6, we can show that ¯ L (t) = 0 for all t ≥ 0. Q ¯ ≡ 0 and it follows that the network is weakly stable. Therefore, Q(t)
2
Lemma 4.8 Under the heavy traffic condition, if m2 +m4 +m6 > 2 then the DHV fluid network is not weakly stable.
28
Proof. Actually, this lemma is implied in part (b) of the proof of Theorem 2.4 in Dai, et al. (1999). We only outline the idea of the proof here and omit the details. Suppose the DHV fluid network is weakly stable under the condition m2 + m4 + m6 > 2. Let Q(t) and T (t) be the queue length process and the time allocation process of the DHV queueing network with Q(0) = 0. Then, from Theorem 4.1 of Chen (1995), 1 ¯ (Q(nt), T (nt)) → (Q(t), T¯(t)), u.o.c., as n → ∞, n
(78)
¯ k (t) ≡ 0 and T¯k (t) ≡ mk t (k = 1, · · · , 6). Hence, with Q T¯2 (t) + T¯4 (t) + T¯6 (t) = (m2 + m4 + m6 )t > 2t (t > 0).
(79)
On the other hand, we know that classes 2, 4 and 6 in the DHV queueing network form a pseudostation (see Hasenbein (1997)), which means that no more than two of these three classes of jobs could be served at the same time. This implies T2 (t) + T4 (t) + T6 (t) ≤ 2t, which, combined with convergence (78), further implies 1 T¯2 (t) + T¯4 (t) + T¯6 (t) = lim (T2 (nt) + T4 (nt) + T6 (nt)) ≤ 2t, n→∞ n for any t ≥ 0. Now, inequalities (79) and (80) contradict each other.
(80) 2
Lemma 4.9 Under the heavy traffic condition, if m2 +m4 +m6 = 2 then the DHV fluid network does not satisfy the SHP -condition. ¯ satisfying equations (16)-(18) Proof. It is sufficient to construct a divergent fluid path Q ¯ with nonzero initial fluid level (i.e., ||Q(0)|| > 0), and it can be verified directly that the following
29
construction suffices:
m3 0 0 µ4 m5 µ4 m3 0
m4 0 0 0 0 µ6 m1
+
1 0 0 −µ4 µ4 − µ5 µ5 − µ6
1 − µ1 µ1 − µ2 µ2 0 0 −µ6
+ ¯ Q(t) = 0 1 µ2 m3 −µ2 µ m µ −µ 2 1 2 3 + µ3 − µ4 0 0 µ4 0 0 Q(t ¯ − [t])
and T¯˙ (t) =
t
0 ≤ t < m5
(t − m5 )
m5 ≤ t < m5 + m1
(t − m5 − m1 ) m5 + m1 ≤ t < m5 + m1 + m3 = 1
t≥1
(0, 0, 0, 1, 1, 1) 0 ≤ t < m5
(1, 1, 0, 0, 0, 1) m5 ≤ t < m5 + m1 (0, 1, 1, 1, 0, 0) m5 + m1 ≤ t < m5 + m1 + m3 = 1 ¯ T˙ (t − [t]) t≥1
¯ ¯ where [t] is the greatest integer that is less than or equal to t. In particular, Q(1) = Q(0).
2
Theorem 4.10 Suppose that the convergence (22) holds. Under the heavy traffic condition, if m2 + m4 + m6 < 2, the weak convergence (31) holds with the limits defined by equations (32)ˆ is a martingale with respect to the filtration generated by (X, ˆ YˆL ). (37). Moreover, the process X n ˆ On the other hand, if m2 + m4 + m6 > 2, ||Q{2,4,6} || diverges to infinity as n → ∞ Proof. The first half of the theorem follows from Proposition 4.4 and Theorem 3.1, and the second half of the theorem follows from Lemma 4.8. 2 Remark 1. In all the examples studied before, the sufficient and necessary conditions for the existence of the diffusion approximation for those priority queueing networks coincide with ˜ L is completely-S. However, the DHV network provides a the condition that the matrix R ˜ L is completely-S, counterexample. In particular, when m2 + m4 + m6 > 2, the matrix R and hence, the “limit processes” in Theorem 3.1 would be well-defined; however, the weak 30
convergence (31) does not hold, i.e., the diffusion approximation does not exist in this case. 2. We have not been able to characterize the diffusion limit for the boundary case m2 + m4 + m6 = 2.
A Counterexample for the Converse of Proposition 3.4 We construct a 9-class variation of the DHV fluid network, and then show that for some parameters, the variation satisfies the SHP -condition but not the L2 -condition. We provide an intuitive presentation, from which it is not difficult to write down a rigorous proof. The variation of the DHV network is as shown in Figure 3. The new network has three additional classes indexed by 7, 8 and 9, which are served by stations 1, 2 and 3, respectively. These three additional classes are assigned the lowest priorities at their respective stations. The additional parameters are positive exogenous arrival rates α7 , α8 and α9 , and the positive mean service times m7 , m8 and m9 . All the other settings of this fluid network are the same as those of the DHV fluid network.
Station 1 -
m7
α1 = 1-
-
Station 2 -
m8
m1
-
m4
-
-
Station 3 -
m9
m2
-
m3
m5
-
m6
-
-
-
Figure 3: A variation of the DHV fluid network From Theorem 2.5 in Dai, et al. (1999), there exists a mean service time vector m∗ = such that the DHV fluid network is stable, but there does not exist a linear Lyapunov function for its fluid level process. Now, let m∗7 = 1−m∗1 −m∗4 , m∗8 = 1−m∗2 −m∗5 , m∗9 = 1−m∗3 −m∗6 and αi = 1 (i = 7, 8, 9). We claim that with the these parameters, the 9-class variation of the DHV network satisfies the SHP -condition but not the L2 -condition. This is because the three additional classes have the lowest priorities at their respective stations and the higher priority classes 1, ..., 6 behave the same as they are in the original DHV network. In addition, for the ˜ L = diag(µ∗ , µ∗ , µ∗ ), and hence is completely-S. 9-class network, the matrix R 7 8 9 Therefore, this variation of the DHV network provides a counterexample to the converse of Proposition 3.4. (m∗1 , ..., m∗6 )
31
5
Concluding Remarks
In this paper, we establish a sufficient condition for the diffusion approximation of multiclass queueing networks under priority service disciplines. This sufficient condition is weaker than the sufficient condition given by Chen and Zhang (2000b), and is more explicit (in terms of network parameters) than the condition given by Bramson (1998). With this sufficient condition, we establish the diffusion approximation for the reentrant line network under the LBFS discipline. By studying a three-station DHV network, we observe that the diffusion approximation may not ˜L exist, even if the “proposed” limiting SRBM exists (i.e., the corresponding reflection matrix R is completely-S).
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