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Lyapunov method for the stability of fluid networks
Operations Research Letters 28 (2001) 125–136
www.elsevier.com/locate/dsw
Lyapunov method for the stability of $uid networks Heng Qing Yea; 1 , Hong Chenb; ∗; 2 b Faculty
a Faculty of Business Administration, National University of Singapore, Singapore of Commerce and Business Administration, University of British Columbia, 2053 Main Mall Vancouver, BC, Canada V6T 1Z2
Received 1 May 2000; received in revised form 1 December 2000; accepted 1 January 2001
Abstract One of the primary tools in establishing the stability of a $uid network is to construct a Lyapunov function. In this paper, we establish the su1ciency in the use of a Lyapunov function. Speci3cally, we show that a necessary and su1cient condition for the stability of a generic $uid network is the existence of a Lyapunov function for its $uid level process. Then by applying this result to various speci3c $uid networks, including a $uid network under all work-conserving service disciplines, a $uid network under a priority service discipline, and a $uid network under a 3rst-in-3rst-out service discipline, we establish the existence of a Lyapunov function for their $uid level processes is a necessary and su1cient condition for their stabilities. c 2001 Elsevier Science B.V. All The result is also applied to various $uid limit models and a linear Skorohod problem. rights reserved. Keywords: Queueing network; Fluid network; Stability; Lyapunov method; Linear Skorohod problem
1. Introduction The phenomenon of queue exists widely in many aspects of manufacturing, telecommunication and service systems. The queueing network theory aims to evaluate, control and design various kinds of queueing network systems. The stability issue arises 3rst from ∗
Corresponding author. Tel.: +1-604-822-8360; fax: +1-604822-9574. E-mail address: [email protected] (H. Chen). 1 Part of this work was done while the author was a PhD Student in the Hong Kong University of Science and Technology. Supported in part by a grant from the Academic Research Fund (Singapore). 2 Supported in part by a grant from NSERC (Canada) and a grant from RGC (Hong Kong).
the study of control of semiconductor fabrication process, which is modeled by a special class of queueing networks, known as reentrant lines. Traditionally, it was believed that, a queueing network would be stable (roughly speaking, the limiting distribution of the queue length process is 3nite) if the tra1c intensity at each station is less than one. Kumar and Seidman [23] provide a two-station priority network as a counterexample to this common belief. Subsequently, a number of other surprising counterexamples appeared; see Lu and Kumar [24], Rybko and Stolyar [27], Bramson [4] and Seidman [28]. These counterexamples have attracted a lot of attention to the study of the stability of queueing networks. An important tool for studying the stability of a queueing network is its corresponding $uid network,
c 2001 Elsevier Science B.V. All rights reserved. 0167-6377/01/$ - see front matter PII: S 0 1 6 7 - 6 3 7 7 ( 0 1 ) 0 0 0 6 0 - 8
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which is a continuous analog of the queueing network. An elegant theorem proposed by Rybko and Stolyar [27] and extended by Dai [13] states that a queueing network is stable if its corresponding $uid limit model or $uid network model is stable. This motivates the study of the stability of $uid networks. The use of Lyapunov (direct) method to study the stability of the $uid network is inspired by its successful application to the (deterministic) dynamic system. To understand the Lyapunov method, we outline here a version of the Lyapunov method for the stability of a dynamic system in an informal way. Consider an ordinary diKerential equation dx = f(t; x); (1) dt where x(·) : R+ → Rn , f(·; ·) : R+ × Rn → Rn . The solution x ≡ 0 is said to be uniformly asymptotically stable if there exists a number ¿0 such that for any solution x(·) of (1) under initial condition x(0)¡ (where the norm · is de3ned in (2) below), lim x(t) = 0:
t→+∞
Under some mild technical condition, the solution x ≡ 0 is uniformly asymptotically stable if and only if there exists a Lyapunov function. (We shall provide a formal de3nition of the Lyapunov function in the next section.) The above necessary condition is also referred to as the converse theorem. Readers may refer to any graduate textbooks on ordinary diKerential equation theory (e.g., Miller and Michel [26]). For establishing the stability of $uid networks, almost all the previous work makes use of some speci3c forms of Lyapunov functions. (Some exceptions include Bertsimas et al. [2], Chen and Zhang [11], and Winograd and Kumar [30].) For example, quadratic Lyapunov functions are used in Kumar and Meyn [22] and Chen [8] for networks under general work-conserving service disciplines; piecewise linear Lyapunov functions are used in Botvich and Zamyatim [3], Down and Meyn [18], Dai and Weiss [17], Dai and Vande Vate [16], Dai et al. [14] and Chen and Ye [10] for networks under general work-conserving disciplines or priority service discipline; linear Lyapunov functions are used in Chen [9] for a linear Skorohod problem and in Chen and Zhang [12] for networks under priority service disciplines; and entropy functions are used in Bramson [5,6] for Kelly
type networks under 3rst-in-3rst-out (FIFO) service disciplines and for networks under head-of-the-line proportional processor sharing disciplines. The basic idea of the use of a Lyapunov function to study the stability of a $uid network can be outlined as follows. A $uid network is stable if its $uid level process Q, a multi-dimensional nonnegative function on [0; ∞), with a unit initial $uid level is drained to zero within a 3xed 3nite time. To establish the stability, it is su1cient to 3nd a nonnegative real function f(·) on [0; ∞) and an ¿0 such that, f(t) = 0 ⇔ Q(t) = 0; and df(t) ¡− dt
if Q(t)¿0;
for almost all t ¿ 0. The key to this approach is how to choose the function f. For example, a linear Lyapunov function of the form, f(t) = h Q(t)
(with constant vector h¿0);
is used in Chen and Zhang [12] to study the stability of a multiclass $uid network under a priority service discipline. However, as far as we know, no research has been carried out on the reverse side, i.e., whether a Lyapunov function always exists when the $uid network is stable. In this paper, we establish the converse theorem in the context of various $uid network models. This paper is organized as follows. In the next section, we propose a Lyapunov approach to the stability of a generic $uid network (GFN), and establish a necessary and su1cient condition for its stability. Establishing this condition is the main result of this paper. This result is then applied to $uid networks under various service disciplines in Section 3, to $uid limit networks (of queueing networks) under various service disciplines in Section 4, and to a linear Skorohod problem in Section 5. We conclude in Section 6. Finally, we introduce some notation and convention that are used throughout this paper. The J -dimensional Euclidean space is denoted by RJ , and its nonnegative J 1 orthant by R+ . Let R = R1 and R+ = R+ . Vectors are understood to be column vectors. The transpose of a vector or a matrix is obtained by adding a prime to it. The notation I represents an identity matrix and e represents a vector with all elements equal to one. The
H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
dimensions of I and e can be easily deduced from the context. Let x be a J -dimensional vector; then diag(x) denotes a J × J diagonal matrix with the jth diagonal element equal to xj . For any x ∈ RJ , the norm is de3ned to be x =
J
|xj |:
(2)
j=1
(The speci3c norm is not essential; in other words, all the results still hold if any other norms are used.) To present our results conveniently, we introduce the continuous function space C J (R+ ), which is the space of all continuous functions f : R+ → RJ . A sequence {fn (t)} of continuous functions in C J [0; ∞] is said to converge uniformly on compact set (u.o.c.) to a continuous function f(t) ∈ C J (R+ ), if for any T ¿0, lim
sup fn (t) − f(t) = 0;
n→∞ 06t6T
and we denote it by fn → f;
u:o:c: as n → ∞:
2. The GFN and its stability First we introduce the GFN model. Definition 2.1. A set of functions Q(·) : R+ → K is said to be a GFN model, if the following three R+ conditions are satis3ed: (a) (Lipschitz condition) There exists an M such that, for any Q(·) ∈ and t1 ; t2 ∈ R+ , Q(t1 ) − Q(t2 ) 6 M |t1 − t2 |: (b) (Scale property) Q(·) ∈ implies (1=r)Q(r·) ∈ for any 3xed r¿0. (c) (Shift property) Q(·) ∈ implies Q(s + ·) ∈ for any 3xed s ¿ 0. Furthermore, if the following condition is also satis3ed, then we call a closed GFN model. (d) (Closeness property) If {Qn }∞ n=1 ⊂ converges to Q u.o.c., then Q ∈ . We call M the Lipschitz constant for the GFN model , and call each element Q ∈ a GFN path (of ). Denote the set of GFN path with total initial level x(¿ 0) as (x) := {Q ∈ : Q(0) = x}:
127
The framework of a GFN model captures some important properties of most well-known $uid networks. As we shall see in the next two sections, a GFN path Q(t) in this de3nition may correspond to a $uid level process in a $uid network or a $uid limit model. The stability of a GFN model is de3ned as follows. Definition 2.2. A GFN model is said to be stable if there exists a time ¿0 such that Q( + ·) ≡ 0 for any GFN path Q(·) ∈ (1). In order to state our main result, we introduce two conditions for a GFN model in the following. The 3rst one is a Lyapunov condition and the second one is based on Stolyar [29]. L-condition: There exist strictly increasing continuous functions wi : R+ → R+ with wi (0) = 0 (i = 1; 2; 3), such that for any GFN path Q ∈ , there is an absolutely continuous function v(t) such that w1 (Q(t)) 6 v(t) 6 w2 (Q(t));
(3)
v(t) ˙ 6 −w3 (Q(t));
(4)
for almost all t ¿ 0. (The function v is often called a Lyapunov function for Q.) S-condition: For any GFN path Q(t) ∈ (1), inf Q(t)¡1:
t¿0
The following main theorem describes the Lyapunov method (including the converse theorem) in the context of a GFN model. Theorem 2.3. (i) A GFN model is stable if and only if the L-condition is satis9ed. (ii) A closed GFN model is stable if and only if one of the following two conditions hold: (a) the L-condition; (b) the S-condition. In fact; for either (i) or (ii); we can choose ∞ v(t) = Q(s) ds: t
Proof. Part (i): (Su
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Step 1. Prove that if Q ∈ , then limt→∞ Q(t) = 0. We further divide the proof of this statement into two substeps. Step 1.1. Prove limt→∞ v(t) = 0 3rst. According to the L-condition, v(t) is a nonnegative (v(t) ¿ w1 (Q(t)) ¿ 0) and nonincreasing (v(t) ˙ 6 −w3 (Q(t)) 6 0 a:e:) function. Hence, there exists a real number c ¿ 0, such that limt→∞ v(t) = c. Next, we show that c = 0 by contradiction. Suppose c¿0. Then
Let Q ∈ (1). Since Q(t) → 0, there exists a time s¿0 such that Q(s) = r and Q(t)¿r for all t ∈ [0; s). According to the L-condition, we have v(t) ˙ 6 −w3 (Q(t)) 6 −w3 (r) for all t ∈ [0; s). Then s 0 6 v(s) = v(0) + v(t) ˙ dt 6 w2 (Q(0)) −
w2 (Q(t)) ¿ v(t) ¿ c; and therefore, by the de3nition of the functions w2 ; w3 , Q(t) ¿
! := w2−1 (c)¿0;
w3 (Q(t)) ¿ m := w3 (!)¿0:
v(t) − v(0) =
0
6
t
0
t
v(s) ˙ ds 6
0
w3 (r) dt = w2 (1) − w3 (r)s;
0
and s 6 w2 (1)=w3 (r):
min{t ¿ 0: Q(t) = 0} 6 : t
Using the shift and scale properties and inequality (5), it can be veri3ed directly that, for any given GFN path Q ∈ (1), there is a sequence of time 0 = 0 ¡1 ¡2 ¡ · · · such that
−w3 (Q(s)) ds
−m ds = −mt:
Q(m ) = r m
This leads to v(t) → −∞ as t → +∞, contradicting v(t) ¿ c¿0 for all t ¿ 0. Step 1.2. Prove limt→∞ Q(t) = 0. Let be any real positive number. Since v(t) → 0 (as t → ∞) and w1 ( )¿0, there is a time t0 , such that v(t) 6 w1 ( ) for all t ¿ t0 . Then for all t ¿ t0 , w1 (Q(t)) 6 v(t) 6 w1 ( );
m − m−1 6 r m−1 t
and
for m = 1; 2; : : : : Then, we have ∗ := lim m 6 m→∞
∞
t r m−1 = :=
m=1
t ¡∞; 1−r
and hence,
Q(∗ ) = Q lim m = lim Q(m ) m→∞
and hence, by the increasing property of w1 ,
m→∞
m
= lim r = 0: m→∞
Q(t) 6 : Step 2. Show that there is a time such that, for any GFN path Q with Q(0) = 1 (i.e., Q ∈ (1)), we have Q(t) = 0 for all t ¿ . We divide the proof into three substeps. Step 2.1. Let 0¡r¡1. Show that there exists a time t ∈ [0; ∞) such that, for all Q ∈ (1), min{t ¿ 0: Q(t) = r} 6 t :
s
Therefore, t = w2 (1)=w3 (r) satis3es (5). Step 2.2. Show that there exists a ∈ [0; ∞) such that, for all Q ∈ (1),
and then, for all t ¿ 0,
Now,
0
(5)
Step 2.3. Show that if Q(∗ )=0, then Q(∗ +·) ≡ 0. For any t ¿ ∗ , we have 0 6 w1 (Q(t)) 6 v(t) 6 v(∗ ) 6 w2 (Q(∗ )) =w2 (0) = 0; and hence Q(t) = 0: (Necessity) Next, we prove that the L-condition is necessary. Let M ¿0 be the Lipschitz constant for the
H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
GFN model . De3ne w1 ; w2 ; w3 as follows: 2
x ; w2 (x) := x2 (1 + M); 2M for all x ¿ 0. Let Q ∈ . Let ∞ v(t) = Q(s) ds: w1 (x) :=
w3 (x) := x;
t
Now, we verify (3) and (4). Since Q satis3es the Lipschitz condition with Lipschitz constant M , we have Q(s) ¿ Q(t) − M (s − t) and then, ∞ Q(s) ds ¿ v(t) = t
¿
for any s ¿ t;
t+(Q(t)=M )
t
t+(Q(t)=M )
t
Q(s) ds
[Q(t) − M (s − t)] ds
2
for all s ¿ t + Q(t):
For s ∈ [t; t + Q(t)], we have, by the Lipschitz condition, Q(s) 6 Q(t) + M Q(t):
v(t) =
t
6
∞
Q(s) ds =
t+Q(t)
t
= Q(t) (1 + M) = w2 (Q(t)):
(6)
Otherwise, for a sequence {rn } ⊂ (0; 1) satisfying limn→∞ rn = 1, there exists a sequence {Qn } ⊂ (1) such that inf Qn (t) ¿ rn :
Then, we can choose a subsequence {ni } ⊂ {1; 2; : : :} such that u:o:c: as ni → ∞;
t
t+Q(t)
Q(t) = lim Qni (t) ¿ lim rni = 1: ni →∞
ni →∞
This contradicts the S-condition. Next, we show by contradiction again that there exists a time t ∈ [0; ∞) such that min{t ¿ 0: Q(t) = r} 6 t
min{t ¿ 0: Qn (t) = r}¿tn : Then, we can again choose a subsequence {ni } ⊂ {1; 2; : : :} such that Qni → Q ∈ (1);
Q(s) ds
Q(t)(1 + M) ds
2
inf Q(t)¡r:
t¿0
holds for all Q ∈ (1). Otherwise, for a sequence {tn } ⊂ [0; ∞) with limn→∞ tn = ∞, there exists a sequence {Qn } ⊂ (1) such that
and this directly implies
Part (ii). The proof of the fact that the S-condition implies the stability of the closed GFN model is almost a word for word repetition of the proof of Theorem 6:1 in Stolyar [29]. We outline the proof in the following three steps. First, we show by contradiction that there exists an r ∈ (0; 1) such that, for all Q ∈ (1),
and hence, for any 3xed t ¿ 0,
Q∗ ( + ·) ≡ 0;
Then,
v(t) ˙ = −Q(t) = w3 (Q(t)):
Qni → Q ∈ (1);
Next, we verify the other half of condition (3). Let t be 3xed and q = Q(t) at the moment. Suppose q¿0 for convenience since q = 0 is a trivial case. Then, by the scale and shift property, Q∗ (s) = (1=q)Q(t + qs) is a GFN path in (1). Since the GFN model is stable, we have
Q(s) = 0
Finally, we have
t¿0
Q(t) 2M = w1 (Q(t)):
=
129
u:o:c: as ni → ∞:
Now, for any 3xed t ¿ 0, we have ni Q(t) = lim Q (t) ¿r ni →∞ since Qn (t) ¿ r for su1ciently large n. This contradicts inequality (6). Finally, we can 3nish proving that the S-condition implies the stability of the closed GFN model by repeating steps 2.2 and 2.3 in the proof for part (i) (su1ciency).
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It follows immediately from the part (i) that stability of the closed GFN model implies the L-condition, and it is obvious that the L-condition implies the S-condition by their de3nitions. Hence, the stability, the L-condition and the S-condition of the closed GFN model are equivalent. 3. Applications to some fluid networks A $uid network is a continuous analog of its corresponding queueing network. It can be described by using a system of equations. The importance of the $uid network in studying the stability of the corresponding queueing network lies in the fact that the stability of the $uid network is a su1cient condition for the stability of the corresponding queueing network. In this section, we show that the main theorem in the previous section can be applied to $uid networks under general work-conserving, priority, and FIFO service disciplines. 3.1. A Fluid network under general work-conserving service disciplines The $uid network model consists of J stations, indexed by j = 1; : : : ; J . Each station has an in3nite storage capacity, and stations are interconnected by frictionless pipes to form a network in which several classes of $uids simultaneously circulate. There are K classes of $uids, indexed by k = 1; : : : ; K, and we denote the set of classes by K = {1; : : : ; K}. Class k $uid resides exclusively in station %(k), where %(·) is a many-to-one mapping from the set of classes K to the set of stations J ≡ {1; : : : ; J }. Let C(j) = {k ∈ K: %(k) = j} be the set of classes that reside in station j. Let C = (cjk )J ×K be a J × K constituent matrix, with cjk = 1 if %(k) = j, and cjk = 0 otherwise. To avoid triviality, we assume that C(j) is nonempty for all j = 1; : : : ; J , i.e. all stations must be resided by at least one class of $uids. To describe the $uid network model, we take as primitives two K-dimensional nonnegative vectors Q(0) = (Qk (0)) and & = (&k ), one K-dimensional positive vector ' = ('k ), one K × K substochastic matrix P = (p‘k )K×K with a spectral radius strictly less than one, and one J × K constituent matrix C. For k = 1; : : : ; K, we interpret the kth component Qk (0)
of Q(0) as the initial $uid level of class k (at station %(k)), the kth component &k of & as the exogenous in$ow rate of class k $uid, and the kth component 'k of ' as the potential out$ow rate of class k $uid, or the processing (or service) rate for class k $uid. Namely, if we assume that the time that can be allocated to processing or serving class k $uid during the time interval [s; t] with s 6 t, is t − s, then 'k (t − s) is the maximum possible amount of out$ow of class k $uid in this duration. The (‘; k)th element p‘k of P represents the proportion of the out$ow of Kclass ‘ $uid that turns into class k $uid and 1 − k=1 p‘k is the fraction that leaves the $uid network. Thus, the condition that P has a spectral radius strictly less than one implies that all $uids will visit only a 3nite number of stations before leaving the $uid network. A $uid network model described by the above primitives is referred to as the $uid network (&; '; P; C) with initial $uid level Q(0). Vectors Q(0), & and ' are referred to as the initial >uid level (vector), (exogenous) in>ow rate (vector), and processing rate (vector), respectively. Matrix P is referred to as the >ow transfer matrix. To be consistent with the corresponding terms used for describing a queueing network, sometimes we also call &, ' and P the (exogenous) arrival rate (vector), service rate (vector), and routing matrix, respectively. The processes that conveniently describe the performance of the $uid network are the K-dimensional >uid level process Q = {Q(t); t ¿ 0} and the K-dimensional allocation process T = {T (t); t ¿ 0}. The kth component of T (t), Tk (t), denotes the total amount of time that station %(k) has devoted to processing class k $uid during time interval [0; t]. The dynamics of the work-conserving $uid network can be summarized as follows: Q(t) = Q(0) + &t − (I − P )DT (t) ¿ 0;
(7)
T (·) is nondecreasing with T (0) = 0;
(8)
I (t) = et − CT (t) is nondecreasing; ∞ (CQ(t)) dI (t) = 0;
(9)
0
(10)
where D = diag('). The J -dimensional process I (t) is referred to as the unused capacity process, whose jth component, Ij (t), denotes the (cumulative) unused
H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
capacity of station j during the time interval [0; t] after processing all classes at station j. The relation (7) is the $ow balance relation; its kth coordinate reads as, Qk (t) = Qk (0) + &k t + −'k Tk (t) ¿ 0;
K
Qkn → Uj (u:o:c:), there exists N , for any n¿N , we have k∈C( j)
|Ujn (s) − Uj (s)|¡
on [t − ; t + ];
p‘k '‘ T‘ (t)
and hence
k = 1; : : : ; K:
Ujn (s) ¿ Uj (s) − ¿ ¿0:
‘=1
Eq. (10) is the work-conserving condition. Note that Q ¿ 0 and dI ¿ 0; then Eq. (10) implies that (CQ(t)) dI (t) = 0 and further each term in this summation, Qk (t) dIj (t) = 0; j ∈ J: k∈C( j)
This means that there could be unused capacity at station j only when $uid levels of all classes processed at station j are zero. Readers are referred to Chen [8] for more detailed description of the $uid network with general work-conserving service disciplines. Any pair (Q; T ) that satis3es Eqs. (7) – (10) is said to be a >uid solution of the work-conserving $uid network. Let ,C denote the set of all $uid solutions (Q; T ) of the work-conserving $uid network (7) – (10) and C = {Q: ∃T such that (Q; T ) ∈ ,C }
131
Then Ijn (s) = Ijn (t − )
for all s ∈ [t − ; t + ]:
Letting n → ∞ in the above yields, Ij (s) = Ij (t − )
for all s ∈ [t − ; t + ];
which immediately implies I˙j (t) = 0. Lemma 3.2. Suppose (Q; T ) ∈ ,C is a >uid solution. Then for any given time s ¿ 0 and any given scaler r¿0; (Q(s + ·); T (s + ·) − T (s)) and ((1=r)Q(r·); (1=r)T (r·)) are also in ,C . ˜ Proof. It can be veri3ed directly that both (Q(t); ˆ T˜ (t)) = (Q(s + t); T (s + t) − T (s)) and (Q(t); Tˆ (t)) = ((1=r)Q(rt); (1=r)T (rt)) satisfy the relations (7) – (10). Lemma 3.3. C is a closed GFN model.
be the set of all feasible $uid level processes. The work-conserving $uid network is said to be stable if there is a time ¿ 0 such that Q( + ·) ≡ 0 for any Q ∈ C with Q(0) = 1.
Proof. The Lipschitz property can be seen from conditions (7) – (9). It can also be veri3ed the closeness, scale and shift properties directly from the de3nition of C and Lemmas 3.1 and 3.2.
Lemma 3.1. Suppose (Qn ; T n ) ∈ ,C (n = 1; 2; : : :) is a sequence of >uid solutions; and
Lemma 3.3, combined with Theorem 2.3, leads to the following theorem on a necessary and su1cient condition for the stability of the $uid network under general work-conserving service disciplines.
(Qn ; T n ) → (Q; T );
u:o:c: as n → ∞
with Q(0)¡∞. Then; (Q; T ) ∈ ,C ; i.e.; (Q; T ) is also a >uid solution. Proof. First, it can be checked directly that (Q(t); T (t)) satis3es (7) – (9). To verify (10), it is su1cient to prove that, I˙j (t) = 0 if k∈C( j) Qk (t)¿0. It can be proven by using the continuity of the one dimensional re$ection mapping (see for example Harrison [20]); however, we include a more direct proof here. Suppose Uj (t) ≡ k∈C( j) Qk (t)¿0 for some station index j. Then, there exist ; ¿0, such that Uj (s) ¿ 2 for any s ∈ [t − ; t + ]. Since Ujn ≡
Theorem 3.4. The work-conserving >uid network (7)–(10) is stable if and only if one of the following two conditions holds: (a) the L-condition is satis9ed for C ; (b) the S-condition is satis9ed for C . 3.2. A >uid network under a priority service discipline Relations (7) – (10) that de3ne the K-dimensional $uid level process Q and K-dimensional allocation
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H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
process T essentially contain K + J equations: Kdimensional balance Eq. (7) and J -dimensional work-conserving condition (10). The other (K − J )-dimensional freedom is usually determined by the choice of some speci3c service disciplines. In this section, we specify a priority service discipline. Let 1 be a one-to-one mapping from K onto K. For any given ‘ and k, if 1(‘)¡1(k) and %(‘) = %(k), then class k $uid does not get processed at station %(k) unless the $uid level of class ‘ equals zero. In short, we say that class ‘ has a higher priority than class k. For convenience, the mapping 1 is often expressed as a permutation of K, i.e., which can be written as 1 = (i1 ; : : : ; iK ) if 1(k) = ik , k ∈ K. To describe the $uid network under the priority discipline, we introduce a K-dimensional unused capacity process Y = {Y (t); t ¿ 0}, where Yk (t), the kth component of Y (t), denotes the (cumulative) unused capacity of station %(k) during the time interval [0; t] after serving all classes at station %(k) which have priority no less than class k (including class k). The quantity Yk (t) can also be interpreted as the (cumulative) remaining capacity at station %(k) for processing $uid of classes at station j having a strictly lower priority than class k. Let
remaining capacity (rate) for processing those classes at station %(k) having a strictly lower priority than class k, only when the $uid levels of all classes in Hk (having a priority no less than k) are zero. (Readers are referred to Chen and Zhang [12] for a description of a queueing network under a priority service discipline that provides an explanation for the relations used in the above.) We shall refer to this network as $uid network (&; '; P; C) with initial $uid level Q(0) under priority service discipline 1. For the priority $uid network (11) – (14). A pair (Q; T ) is said to be a >uid solution if they jointly satisfy (11) – (14). Similar to the case of a $uid network under general work-conserving disciplines, let ,P denote the set of all $uid solutions, and P denote the set of all feasible $uid level processes. The priority $uid network is said to be stable if there is a time ¿ 0 such that Q( + ·) ≡ 0 for any Q ∈ P with Q(0) = 1. We state the following lemma, which can be proven in a similar way as Lemmas 3.1–3.3 and hence is omitted.
Hk = {‘: ‘ ∈ C(%(k)); 1(‘) 6 1(k)}
with Q(0)¡∞. Then; (Q; T ) ∈ ,P ; i.e.; (Q; T ) is also a >uid solution. (ii) Suppose (Q; T ) ∈ ,P is a >uid solution. Then for any given time s ¿ 0 and any given scaler r¿0; (Q(s + ·); T (s + ·) − T (s)) and ((1=r)Q(r·); (1=r)T (r·)) are also in ,P . (iii) P is a closed GFN model.
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. Then the dynamics of the network under the priority discipline 1 can be described as follows: Q(t) = Q(0) + &t − (I − P )DT (t) ¿ 0;
(11)
T (·) is nondecreasing with T (0) = 0; Yk (t) = t − T‘ (t) is nondecreasing;
(12)
‘∈Hk
0
∞
Qk (t) dYk (t) = 0;
k ∈ K; (13)
k ∈ K:
(14)
Relations (11) – (14) subsume relations (7) – (10), if we note that Yk (t) = I%(k) (t), when k has the lowest priority at station %(k). In particular, relation (14) speci3es both the work-conserving condition and the priority discipline; in words, for each k, relation (14) means that at any time t, there could be some positive
Lemma 3.5. (i) Suppose (Qn ; T n ) ∈ ,P (n = 1; 2 : : :) is a sequence of >uid solutions; and (Qn ; T n ) → (Q; T );
u:o:c: as n → ∞
Lemma 3.5, combined with Theorem 2.3, implies the following theorem on a necessary and su1cient condition for the stability of a $uid network under a priority service discipline. Theorem 3.6. The priority >uid network (11)–(14) is stable if and only if one of the following two conditions holds: (a) the L-condition is satis9ed for P ; (b) the S-condition is satis9ed for P . 3.3. A >uid network under FIFO service discipline The settings of the FIFO $uid network parallel those of a priority $uid network, but with a diKerent set of
H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
dynamic relations to complement the dynamic relations of a $uid network under general work-conserving disciplines. The parameters (&; '; P; C), the $uid level process Q and the allocation process T are also de3ned in the same way as those for the work-conserving $uid network. The dynamics of the FIFO $uid network can be summarized as follows: (7) – (10), and D(t) = M −1 T (t);
A(t) = &t + P D(t); W (t) = CMQ(t);
(15) t ¿ 0; t ¿ 0;
(16) (17)
Dk (t + Wj (t)) = Qk (0) + Ak (t); t ¿ 0; j = %(k); k = 1; : : : ; K;
(18)
where M = diag(m). The K-dimensional processes A={A(t); t ¿ 0} and D={D(t); t ¿ 0} are referred to as the in>ow (or arrival) process and the out>ow (or departure) process, respectively. The J -dimensional process W (t) is referred to as the workload process. The jth component of W (t), Wj (t), is the amount of time required for station (server) j to process all the $uid presented at station j at time t. Relation (18) is the key relation that speci3es the FIFO service discipline. Readers are referred to Harrison [21] for a more detailed description, and Chen and Zhang [11] for an alternative description. Again, for the FIFO $uid network (7) – (10) and (15) – (18), we also de3ne the $uid solution (Q; T ), the set of $uid solutions ,F , the set of all feasible $uid level processes F , and the stability concept, in the same way as we did for the work-conserving $uid network. Similar to Lemma 3.5, we have the following lemma. Lemma 3.7. (i) Suppose (Qn ; T n ) ∈ ,F (n = 1; 2 : : :) is a sequence of >uid solutions of the >uid network; and (Qn ; T n ) → (Q; T );
u:o:c: as n → ∞:
Then; (Q; T ) ∈ ,F ; i.e.; (Q; T ) is also a >uid solution. (ii) Suppose (Q; T ) ∈ ,F is a >uid solution. Then for any given time s ¿ 0 and any given scaler r¿0; (Q(s + ·); T (s + ·) − T (s)) and ((1=r)Q(r·); (1=r)T (r·)) are also in ,F . (iii) F is a closed GFN model.
133
Proof. Parts (ii) and (iii) can be proved in the same manner as Lemmas 3.2 and 3.3, respectively. For part (i), it is su1cient to verify that (Q; T ) satis3es Eqs. (15) – (18). Verifying (15) – (17) is trivial. It is clear that lim Dkn (t + Wjn (t)) = Dk (t + Wj (t))
n→∞
for all t ¿ 0;
since all the functions involved are (Lipschitz) continuous. (Here we append a superscript n to all the processes corresponding to (Qn ; T n ).) It follows immediately from the above convergence that Eq. (18) also holds for (Q; T ). Lemma 3.7, combined with Theorem 2.3, implies the following theorem that describes a necessary and su1cient condition for the stability of the $uid network under FIFO service discipline. Theorem 3.8. The FIFO >uid network (7)–(10) and (15)–(18) is stable if and only if one of the following two conditions holds: (a) the L-condition is satis9ed for F ; (b) the S-condition is satis9ed for F . 4. Application to fluid limit models of queuing networks In this section, we apply the main result to some $uid limit models, the stability of which also implies the stability of the corresponding queueing networks. Both the $uid network and the $uid limit model are the continuous analog of the corresponding queueing network. The diKerence between them lies in the fact that, the $uid network is described by a set of equations (or conditions), while the $uid limit model arises as the limit of sequences of scaled queueing networks. In general, a $uid limit path (to be de3ned later) in the $uid limit model is also a solution of the corresponding $uid network, but the converse may not be true. Therefore, the stability condition of a $uid limit model may be strictly weaker than the stability condition of a corresponding $uid network. The study of the former can be important in view of the recent counterexamples brought forth by Bramson [7] and Dai et al. [15], where the queueing networks are found stable under some arrival=service time distributions, but the
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corresponding $uid networks are not stable. We conjecture that for these queueing networks, their corresponding $uid limit models are stable. 4.1. Queuing network The queueing network consists of J stations indexed by j = 1; : : : ; J; and K job classes indexed by k = 1; : : : ; K. Let uk = {unk ; n ¿ 1} be the sequence of interarrival time process, and v k = {vnk ; n ¿ 1} be the sequence of service time process, where unk indicates the interarrival time between the (n − 1)st and the nth jobs of class k, and vnk indicates the service time for the nth job of class k, k = 1; : : : ; K. A class k job is served at station %(k), and after its service completion, it may become a class ‘ job with probability K pk‘ and leave the network with probability 1 − ‘=1 pk‘ . Let P = (pk‘ ). Let C = (cjk ) be a J × K matrix whose (j; k)th component cjk =1 if j=%(k) and =0 otherwise. While each station may serve more than one class of jobs, each job is served at one speci3c station (determined by the many-to-one mapping %(·)). To describe the state space of the queueing network precisely, we need to specify the service discipline such as the priority service discipline, and FIFO service discipline. We assume all random variables are de3ned on a probability space (7; F; P) with expectation operator E. Assume that u1 ; : : : ; uK ; v 1 ; : : : ; v K are mutually independent iid sequences;
(19)
mk := Ev1k ¡∞
(20)
for k = 1; : : : ; K:
To avoid triviality, assume E := {k: Eu1k ¡∞} = ∅. Furthermore, assume that for each k ∈ E, there exists some integer nk ¿0 and some function pk (x) ¿ 0 on ∞ [0; ∞) with 0 pk (x) d x¿0, such that P{u1k ¿ x}¿0 for each x¿0;
b nk k u‘ 6 b ¿ pk (x) d x P a6 ‘=1
for each 0 6 a¡b:
(21)
a
(22)
Let & = (&k ) be the exogenous arrival rate (vector) where &k = 1=Eu1k for k ∈ E and &k = 0 for k ∈ E; k = 1; : : : ; K.
With the above primitives, we can de3ne a Markov process X = {X (t); t ¿ 0} to describe the dynamics of the queueing network. Then, a queueing network described above is said to be stable if its underlying Markov process X is positive Harris recurrent. (Readers are referred to Dai [13] for a formal description of the state process X (t), and to Dai [13] and Meyn and Tweedie [25] for a formal de3nition of positive Harris recurrence.) 4.2. Fluid limit model Now, we consider a sequence of queueing networks described in the last subsection. Let Qn : R+ → ZK+ ; n = 1; 2; : : : ; be a sequence of corresponding queue length processes with initial queue length Qn (0), where Z+ is the set of all nonnegative integers. The kth coordinate of Qn (t), Qkn (t), indicates the number of class k jobs in the nth network at time t. For almost all sample paths ! ∈ 7 , any increasing positive real number sequence {rn } with rn → ∞, and any sequence of initial queue lengths Qn (0) with (1=rn )Qn (0) → Q(0), there is a subsequence {rnj } of {rn } such that 1 rnj T Q (rnj t) → Q(t); rn j
u:o:c: as nj → ∞:
T Any limit Q(t) de3ned above is called the ( feasible) T >uid limit path with initial $uid level Q(0) The set of all these $uid limit paths, denoted by L , is said to be the >uid limit model. It is known that under the assumptions (21) – (22), the queueing network is stable if the $uid limit model T is stable (i.e., there exists a ¿ 0 such that Q(+·) ≡ T T 0 for any Q ∈ L with Q(0) = 1) (Theorem 4.2 of Dai [13]). This result indicates that the study of the stability of $uid limit model is important for the study of the stability of the corresponding queueing network. For the $uid limit model L , we have the following property. Proposition 4.1. The >uid limit model L is a GFN model. Proof. We need to verify conditions (a) – (c) in the de3nition of GFN model for L . Firstly, condition (a) follows from Theorem 4:1 in Dai [13].
H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
To verify the scale and shift property, we suppose that Q ∈ L . Then there is ! ∈ 7, an increasing positive real number sequence {rn } with rn → ∞, and a sequence of scaled queue length processes Qn (t) such that 1 n T Q (rn t) → Q(t); rn
u:o:c: on t ∈ [0; ∞):
Let rn = r · rn . Then, we have 1 n 1 1 n 1 Q (rn t) = Q (rn (rt)) → Q(rt); rn r rn r u:o:c: on t ∈ [0; ∞); and this shows that 1r Q(rt) ∈ L . For any 3xed number ¿ 0; 1 n T + t); Q (rn ( + t)) → Q( rn u:o:c: on t ∈ [0; ∞); T + t) ∈ L . and hence Q( However, we could not establish the closeness property for the $uid limit model L . Applying Theorem 2.3 to L yields the following theorem. Theorem 4.2. The >uid limit model L stable if and only if the L-condition holds for L . 5. Application to a linear Skorohod problem In this section, we apply our main result to a linear Skorohod problem (LSP), and obtain a necessary and su1cient condition for the stability of this LSP. The LSP is closely related to the theory of queueing networks. Its stability implies the stability of the corresponding semi-martingale re$ected Brownian motion (SRBM) (see Dupuis and Williams [19]), which arises as the diKusion limit of a queueing network. Readers are referred to Chen [9] and the references there for details on LSP and SRBM. Let R be a J × J matrix and < ∈ RJ . Given X (0) ∈ J R+ , the J -dimensional continuous functions from R+ to RJ , Y and Z, are said to solve the LSP(<; R) with initial state X (0), if they jointly satisfy
X (t) dY (t) = 0:
135
(25)
Let ,S be the set of all solutions (X; Y ) of LSP(<; R), and let S be the set, {X : ∃Y s:t: (X; Y ) ∈ ,S }: A LSP(<; R) is said to be stable if, for any number ¿0 and any X ∈ S with X (0) = 1, there exists a ¿ 0 such that X ( + ·)¡ . To be consistent with the existing de3nition in the literature, the stability of the LSP is de3ned slightly diKerent from that of the $uid networks (requiring X ( + ·)¡ instead of X ( + ·) = 0), but we shall see momentarily that they are actually equivalent (under some conditions). 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 next lemma follows from an oscillation inequality by Bernard and El Kharroubi [1]. Lemma 5.1. Suppose R is completely-S. Then; there is a constant M such that any solution (X; Y ) of the LSP(<; R) is Lipschitz continuous with Lipschitz constant M . Then, with some straightforward arguments, we can establish that S is a closed GFN model if R is completely-S. Applying Theorem 2.3 to this case yields the following theorem. Theorem 5.2. Suppose R is completely-S. Then; the LSP(<; R) is stable if and only if one of the following three conditions holds: (a) there exists ¿ 0 such that X ( + ·) ≡ 0; (b) the L-condition is satis9ed for S ; (c) the S-condition is satis9ed for S . Proof. Since S is a closed GFN model under the condition that R is completely-S, we have, by part (ii) of Theorem 2.3, (a) ⇔ (b) ⇔ (c):
X (t) = X (0) +
(23)
By de3nition of stability of LSP, it is also obvious that
dY (t) ¿ 0;
(24)
(a) ⇒ the LSP(<; R) is stable ⇒ (c):
Y (0) = 0;
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6. Concluding remarks In this paper, we have obtained necessary and su1cient conditions for the stability of various $uid models, $uid limit models and a LSP. This shows that the Lyapunov method would play the same role in the stability of $uid models as it plays in the stability of deterministic dynamic systems. In particular, the “only if” part (the converse theorem) supports the previous eKorts on the use of Lyapunov functions to characterize the stability of $uid networks. References [1] A. Bernard, A. El Kharroubi, RWegulation de processus dans le premier orthant de Rn , Stochastics Stochastics Reports 34 (1991) 149–167. [2] D. Bertsimas, D. Gamarnik, J.N. Tsitsiklis, Stability conditions for multiclass $uid queueing networks, IEEE Trans. Automat. Control 41 (1996) 1618–1631. [3] D.D Botvich, A.A. Zamyatin, Ergodicity of conservative communication networks, Rapport de recherche 1772, INTRA, October, 1992. [4] M. Bramson, Instability of FIFO queueing networks, Ann. Appl. Probab. 4 (1994) 414–431. [5] M. Bramson, Convergence to equilibria for $uid models of FIFO queueing networks, Queueing Systems: Theory Appl. 22 (1996) 5–45. [6] M. Bramson, Convergence to equilibria for $uid models of head-of-the-line proportional processor sharing queueing networks, Queueing Systems: Theory Appl. 23 (1997) 1–26. [7] M. Bramson, A stable queueing network with unstable $uid model, Ann. Appl. Probab. 9 (1999) 818–853. [8] H. Chen, Fluid approximations and stability of multiclass queueing networks: Work-conserving discipline, Ann. Appl. Probab. 5 (1995) 637–655. [9] H. Chen, A su1cient condition for the positive recurrence of a semimartingale re$ecting Brownian motion in an orthant, Ann. Appl. Probab. 6 (1996). [10] H. Chen, H.Q. Ye, Piecewise linear lyapunov function for the stability of priority multiclass queueing networks, Proceedings of the 38th IEEE Conference on Decision and Control, 1999, pp. 931–936. [11] H. Chen, H. Zhang, Stability of multiclass queueing networks under FIFO service disciplines, Math. Oper. Res. 22 (3) (1997) 691–725. [12] H. Chen, H. Zhang, Stability of multiclass queueing networks under priority service disciplines, Oper. Res. 48 (2000). [13] J.G. Dai, On positive Harris recurrence of multiclass queueing networks: a uni3ed approach via $uid models, Ann. Appl. Probab. 5 (1995) 49–77.
[14] J.G. Dai, J.J. Hasenbein, J.H. Vande Vate, Stability of a three-station $uid network, Queueing Systems, Theory Appl. 33 (1999) 293–325. [15] J.G. Dai, J.J. Hasenbein, J.H. Vande Vate, Stability gap between $uid and queueing networks, presented at INFORMS Philadelphia Conference, 1999. [16] J.G. Dai, J. Vander Vate, The stability of two-station multi-type $uid networks, Oper. Res. 48 (2000) 721–744. [17] J.G. Dai, G. Weiss, Stability and instability of $uid models for certain reentrant lines, Math. Oper. Res. 21 (1996) 115– 134. [18] D. Down, S. Meyn, Piecewise linear test functions for stability of queueing networks, Queueing Systems, Theory Appl. 27 (1997) 205 –226. Also in Proceeding of the 33rd Conference on Decision and Control, 1994, pp. 2069 –2074. [19] P. Dupuis, R.J. Williams, Lyapunov functions for semimartingale re$ecting Brownian motions, Ann. Probab. 22 (1994) 680–702. [20] J.M. Harrison, Brownian Motion and Stochastic Flow Systems, Wiley, New York, 1985. [21] J.M. Harrison, Balanced $uid models of multiclass queueing networks: a heavy tra1c conjecture, in: F. Kelly, R. Williams (Eds.), Stochastic networks, IMA Volumes in Mathematics and its Application, Vol. 71, Springer, New York, 1995, pp. 1–20. [22] P.R. Kumar, S. Meyn, Stability of queueing networks and scheduling policies, IEEE Trans. Automat. Control 36 (1995) 1406–1416. [23] P.R. Kumar, T.I. Seidman, Dynamic instabilities and stabilization methods in distributed real-time scheduling of manufacturing systems, IEEE Trans. Automat. Control 35 (1990) 289–298. [24] S.H. Lu, P.R. Kumar, Distributed scheduling based on due dates and buKer priorities, IEEE Trans. Automat. Control 36 (1991) 1406–1416. [25] S. Meyn, R.L. Tweedie, Markov Chain and Stochastic Stability, Springer, London, 1993. [26] R.K. Miller, A.N. Michel, Ordinary DiKerential Equations, Academic Press, New York, 1982. [27] A.N. Rybko, A.L. Stolyar, Ergodicity of stochastic processed describing the operations of open queueing networks, Problemy Peredachi Informatsii 28 (1992) 2–26. [28] T.I. Seidman, First come 3rst server can be unstable, IEEE Trans. Automat. Control 39 (1994) 2166–2171. [29] A.L. Stolyar, On the stability of multiclass queueing network: a relaxed su1cient condition via limiting $uid processes, Markov Process Related Fields 1 (4) (1995) 491–512. [30] G.L. Winograd, P.R. Kumar, The FCFS service discipline: stable network topologies, bounds on tra1c burstiness and delay, and control by regulators, Math. Comput. Modeling (Special Issue on Recent Advances in Discrete Event Systems) 23 (11) (1996) 115–129.
www.elsevier.com/locate/dsw
Lyapunov method for the stability of $uid networks Heng Qing Yea; 1 , Hong Chenb; ∗; 2 b Faculty
a Faculty of Business Administration, National University of Singapore, Singapore of Commerce and Business Administration, University of British Columbia, 2053 Main Mall Vancouver, BC, Canada V6T 1Z2
Received 1 May 2000; received in revised form 1 December 2000; accepted 1 January 2001
Abstract One of the primary tools in establishing the stability of a $uid network is to construct a Lyapunov function. In this paper, we establish the su1ciency in the use of a Lyapunov function. Speci3cally, we show that a necessary and su1cient condition for the stability of a generic $uid network is the existence of a Lyapunov function for its $uid level process. Then by applying this result to various speci3c $uid networks, including a $uid network under all work-conserving service disciplines, a $uid network under a priority service discipline, and a $uid network under a 3rst-in-3rst-out service discipline, we establish the existence of a Lyapunov function for their $uid level processes is a necessary and su1cient condition for their stabilities. c 2001 Elsevier Science B.V. All The result is also applied to various $uid limit models and a linear Skorohod problem. rights reserved. Keywords: Queueing network; Fluid network; Stability; Lyapunov method; Linear Skorohod problem
1. Introduction The phenomenon of queue exists widely in many aspects of manufacturing, telecommunication and service systems. The queueing network theory aims to evaluate, control and design various kinds of queueing network systems. The stability issue arises 3rst from ∗
Corresponding author. Tel.: +1-604-822-8360; fax: +1-604822-9574. E-mail address: [email protected] (H. Chen). 1 Part of this work was done while the author was a PhD Student in the Hong Kong University of Science and Technology. Supported in part by a grant from the Academic Research Fund (Singapore). 2 Supported in part by a grant from NSERC (Canada) and a grant from RGC (Hong Kong).
the study of control of semiconductor fabrication process, which is modeled by a special class of queueing networks, known as reentrant lines. Traditionally, it was believed that, a queueing network would be stable (roughly speaking, the limiting distribution of the queue length process is 3nite) if the tra1c intensity at each station is less than one. Kumar and Seidman [23] provide a two-station priority network as a counterexample to this common belief. Subsequently, a number of other surprising counterexamples appeared; see Lu and Kumar [24], Rybko and Stolyar [27], Bramson [4] and Seidman [28]. These counterexamples have attracted a lot of attention to the study of the stability of queueing networks. An important tool for studying the stability of a queueing network is its corresponding $uid network,
c 2001 Elsevier Science B.V. All rights reserved. 0167-6377/01/$ - see front matter PII: S 0 1 6 7 - 6 3 7 7 ( 0 1 ) 0 0 0 6 0 - 8
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H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
which is a continuous analog of the queueing network. An elegant theorem proposed by Rybko and Stolyar [27] and extended by Dai [13] states that a queueing network is stable if its corresponding $uid limit model or $uid network model is stable. This motivates the study of the stability of $uid networks. The use of Lyapunov (direct) method to study the stability of the $uid network is inspired by its successful application to the (deterministic) dynamic system. To understand the Lyapunov method, we outline here a version of the Lyapunov method for the stability of a dynamic system in an informal way. Consider an ordinary diKerential equation dx = f(t; x); (1) dt where x(·) : R+ → Rn , f(·; ·) : R+ × Rn → Rn . The solution x ≡ 0 is said to be uniformly asymptotically stable if there exists a number ¿0 such that for any solution x(·) of (1) under initial condition x(0)¡ (where the norm · is de3ned in (2) below), lim x(t) = 0:
t→+∞
Under some mild technical condition, the solution x ≡ 0 is uniformly asymptotically stable if and only if there exists a Lyapunov function. (We shall provide a formal de3nition of the Lyapunov function in the next section.) The above necessary condition is also referred to as the converse theorem. Readers may refer to any graduate textbooks on ordinary diKerential equation theory (e.g., Miller and Michel [26]). For establishing the stability of $uid networks, almost all the previous work makes use of some speci3c forms of Lyapunov functions. (Some exceptions include Bertsimas et al. [2], Chen and Zhang [11], and Winograd and Kumar [30].) For example, quadratic Lyapunov functions are used in Kumar and Meyn [22] and Chen [8] for networks under general work-conserving service disciplines; piecewise linear Lyapunov functions are used in Botvich and Zamyatim [3], Down and Meyn [18], Dai and Weiss [17], Dai and Vande Vate [16], Dai et al. [14] and Chen and Ye [10] for networks under general work-conserving disciplines or priority service discipline; linear Lyapunov functions are used in Chen [9] for a linear Skorohod problem and in Chen and Zhang [12] for networks under priority service disciplines; and entropy functions are used in Bramson [5,6] for Kelly
type networks under 3rst-in-3rst-out (FIFO) service disciplines and for networks under head-of-the-line proportional processor sharing disciplines. The basic idea of the use of a Lyapunov function to study the stability of a $uid network can be outlined as follows. A $uid network is stable if its $uid level process Q, a multi-dimensional nonnegative function on [0; ∞), with a unit initial $uid level is drained to zero within a 3xed 3nite time. To establish the stability, it is su1cient to 3nd a nonnegative real function f(·) on [0; ∞) and an ¿0 such that, f(t) = 0 ⇔ Q(t) = 0; and df(t) ¡− dt
if Q(t)¿0;
for almost all t ¿ 0. The key to this approach is how to choose the function f. For example, a linear Lyapunov function of the form, f(t) = h Q(t)
(with constant vector h¿0);
is used in Chen and Zhang [12] to study the stability of a multiclass $uid network under a priority service discipline. However, as far as we know, no research has been carried out on the reverse side, i.e., whether a Lyapunov function always exists when the $uid network is stable. In this paper, we establish the converse theorem in the context of various $uid network models. This paper is organized as follows. In the next section, we propose a Lyapunov approach to the stability of a generic $uid network (GFN), and establish a necessary and su1cient condition for its stability. Establishing this condition is the main result of this paper. This result is then applied to $uid networks under various service disciplines in Section 3, to $uid limit networks (of queueing networks) under various service disciplines in Section 4, and to a linear Skorohod problem in Section 5. We conclude in Section 6. Finally, we introduce some notation and convention that are used throughout this paper. The J -dimensional Euclidean space is denoted by RJ , and its nonnegative J 1 orthant by R+ . Let R = R1 and R+ = R+ . Vectors are understood to be column vectors. The transpose of a vector or a matrix is obtained by adding a prime to it. The notation I represents an identity matrix and e represents a vector with all elements equal to one. The
H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
dimensions of I and e can be easily deduced from the context. Let x be a J -dimensional vector; then diag(x) denotes a J × J diagonal matrix with the jth diagonal element equal to xj . For any x ∈ RJ , the norm is de3ned to be x =
J
|xj |:
(2)
j=1
(The speci3c norm is not essential; in other words, all the results still hold if any other norms are used.) To present our results conveniently, we introduce the continuous function space C J (R+ ), which is the space of all continuous functions f : R+ → RJ . A sequence {fn (t)} of continuous functions in C J [0; ∞] is said to converge uniformly on compact set (u.o.c.) to a continuous function f(t) ∈ C J (R+ ), if for any T ¿0, lim
sup fn (t) − f(t) = 0;
n→∞ 06t6T
and we denote it by fn → f;
u:o:c: as n → ∞:
2. The GFN and its stability First we introduce the GFN model. Definition 2.1. A set of functions Q(·) : R+ → K is said to be a GFN model, if the following three R+ conditions are satis3ed: (a) (Lipschitz condition) There exists an M such that, for any Q(·) ∈ and t1 ; t2 ∈ R+ , Q(t1 ) − Q(t2 ) 6 M |t1 − t2 |: (b) (Scale property) Q(·) ∈ implies (1=r)Q(r·) ∈ for any 3xed r¿0. (c) (Shift property) Q(·) ∈ implies Q(s + ·) ∈ for any 3xed s ¿ 0. Furthermore, if the following condition is also satis3ed, then we call a closed GFN model. (d) (Closeness property) If {Qn }∞ n=1 ⊂ converges to Q u.o.c., then Q ∈ . We call M the Lipschitz constant for the GFN model , and call each element Q ∈ a GFN path (of ). Denote the set of GFN path with total initial level x(¿ 0) as (x) := {Q ∈ : Q(0) = x}:
127
The framework of a GFN model captures some important properties of most well-known $uid networks. As we shall see in the next two sections, a GFN path Q(t) in this de3nition may correspond to a $uid level process in a $uid network or a $uid limit model. The stability of a GFN model is de3ned as follows. Definition 2.2. A GFN model is said to be stable if there exists a time ¿0 such that Q( + ·) ≡ 0 for any GFN path Q(·) ∈ (1). In order to state our main result, we introduce two conditions for a GFN model in the following. The 3rst one is a Lyapunov condition and the second one is based on Stolyar [29]. L-condition: There exist strictly increasing continuous functions wi : R+ → R+ with wi (0) = 0 (i = 1; 2; 3), such that for any GFN path Q ∈ , there is an absolutely continuous function v(t) such that w1 (Q(t)) 6 v(t) 6 w2 (Q(t));
(3)
v(t) ˙ 6 −w3 (Q(t));
(4)
for almost all t ¿ 0. (The function v is often called a Lyapunov function for Q.) S-condition: For any GFN path Q(t) ∈ (1), inf Q(t)¡1:
t¿0
The following main theorem describes the Lyapunov method (including the converse theorem) in the context of a GFN model. Theorem 2.3. (i) A GFN model is stable if and only if the L-condition is satis9ed. (ii) A closed GFN model is stable if and only if one of the following two conditions hold: (a) the L-condition; (b) the S-condition. In fact; for either (i) or (ii); we can choose ∞ v(t) = Q(s) ds: t
Proof. Part (i): (Su
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H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
Step 1. Prove that if Q ∈ , then limt→∞ Q(t) = 0. We further divide the proof of this statement into two substeps. Step 1.1. Prove limt→∞ v(t) = 0 3rst. According to the L-condition, v(t) is a nonnegative (v(t) ¿ w1 (Q(t)) ¿ 0) and nonincreasing (v(t) ˙ 6 −w3 (Q(t)) 6 0 a:e:) function. Hence, there exists a real number c ¿ 0, such that limt→∞ v(t) = c. Next, we show that c = 0 by contradiction. Suppose c¿0. Then
Let Q ∈ (1). Since Q(t) → 0, there exists a time s¿0 such that Q(s) = r and Q(t)¿r for all t ∈ [0; s). According to the L-condition, we have v(t) ˙ 6 −w3 (Q(t)) 6 −w3 (r) for all t ∈ [0; s). Then s 0 6 v(s) = v(0) + v(t) ˙ dt 6 w2 (Q(0)) −
w2 (Q(t)) ¿ v(t) ¿ c; and therefore, by the de3nition of the functions w2 ; w3 , Q(t) ¿
! := w2−1 (c)¿0;
w3 (Q(t)) ¿ m := w3 (!)¿0:
v(t) − v(0) =
0
6
t
0
t
v(s) ˙ ds 6
0
w3 (r) dt = w2 (1) − w3 (r)s;
0
and s 6 w2 (1)=w3 (r):
min{t ¿ 0: Q(t) = 0} 6 : t
Using the shift and scale properties and inequality (5), it can be veri3ed directly that, for any given GFN path Q ∈ (1), there is a sequence of time 0 = 0 ¡1 ¡2 ¡ · · · such that
−w3 (Q(s)) ds
−m ds = −mt:
Q(m ) = r m
This leads to v(t) → −∞ as t → +∞, contradicting v(t) ¿ c¿0 for all t ¿ 0. Step 1.2. Prove limt→∞ Q(t) = 0. Let be any real positive number. Since v(t) → 0 (as t → ∞) and w1 ( )¿0, there is a time t0 , such that v(t) 6 w1 ( ) for all t ¿ t0 . Then for all t ¿ t0 , w1 (Q(t)) 6 v(t) 6 w1 ( );
m − m−1 6 r m−1 t
and
for m = 1; 2; : : : : Then, we have ∗ := lim m 6 m→∞
∞
t r m−1 = :=
m=1
t ¡∞; 1−r
and hence,
Q(∗ ) = Q lim m = lim Q(m ) m→∞
and hence, by the increasing property of w1 ,
m→∞
m
= lim r = 0: m→∞
Q(t) 6 : Step 2. Show that there is a time such that, for any GFN path Q with Q(0) = 1 (i.e., Q ∈ (1)), we have Q(t) = 0 for all t ¿ . We divide the proof into three substeps. Step 2.1. Let 0¡r¡1. Show that there exists a time t ∈ [0; ∞) such that, for all Q ∈ (1), min{t ¿ 0: Q(t) = r} 6 t :
s
Therefore, t = w2 (1)=w3 (r) satis3es (5). Step 2.2. Show that there exists a ∈ [0; ∞) such that, for all Q ∈ (1),
and then, for all t ¿ 0,
Now,
0
(5)
Step 2.3. Show that if Q(∗ )=0, then Q(∗ +·) ≡ 0. For any t ¿ ∗ , we have 0 6 w1 (Q(t)) 6 v(t) 6 v(∗ ) 6 w2 (Q(∗ )) =w2 (0) = 0; and hence Q(t) = 0: (Necessity) Next, we prove that the L-condition is necessary. Let M ¿0 be the Lipschitz constant for the
H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
GFN model . De3ne w1 ; w2 ; w3 as follows: 2
x ; w2 (x) := x2 (1 + M); 2M for all x ¿ 0. Let Q ∈ . Let ∞ v(t) = Q(s) ds: w1 (x) :=
w3 (x) := x;
t
Now, we verify (3) and (4). Since Q satis3es the Lipschitz condition with Lipschitz constant M , we have Q(s) ¿ Q(t) − M (s − t) and then, ∞ Q(s) ds ¿ v(t) = t
¿
for any s ¿ t;
t+(Q(t)=M )
t
t+(Q(t)=M )
t
Q(s) ds
[Q(t) − M (s − t)] ds
2
for all s ¿ t + Q(t):
For s ∈ [t; t + Q(t)], we have, by the Lipschitz condition, Q(s) 6 Q(t) + M Q(t):
v(t) =
t
6
∞
Q(s) ds =
t+Q(t)
t
= Q(t) (1 + M) = w2 (Q(t)):
(6)
Otherwise, for a sequence {rn } ⊂ (0; 1) satisfying limn→∞ rn = 1, there exists a sequence {Qn } ⊂ (1) such that inf Qn (t) ¿ rn :
Then, we can choose a subsequence {ni } ⊂ {1; 2; : : :} such that u:o:c: as ni → ∞;
t
t+Q(t)
Q(t) = lim Qni (t) ¿ lim rni = 1: ni →∞
ni →∞
This contradicts the S-condition. Next, we show by contradiction again that there exists a time t ∈ [0; ∞) such that min{t ¿ 0: Q(t) = r} 6 t
min{t ¿ 0: Qn (t) = r}¿tn : Then, we can again choose a subsequence {ni } ⊂ {1; 2; : : :} such that Qni → Q ∈ (1);
Q(s) ds
Q(t)(1 + M) ds
2
inf Q(t)¡r:
t¿0
holds for all Q ∈ (1). Otherwise, for a sequence {tn } ⊂ [0; ∞) with limn→∞ tn = ∞, there exists a sequence {Qn } ⊂ (1) such that
and this directly implies
Part (ii). The proof of the fact that the S-condition implies the stability of the closed GFN model is almost a word for word repetition of the proof of Theorem 6:1 in Stolyar [29]. We outline the proof in the following three steps. First, we show by contradiction that there exists an r ∈ (0; 1) such that, for all Q ∈ (1),
and hence, for any 3xed t ¿ 0,
Q∗ ( + ·) ≡ 0;
Then,
v(t) ˙ = −Q(t) = w3 (Q(t)):
Qni → Q ∈ (1);
Next, we verify the other half of condition (3). Let t be 3xed and q = Q(t) at the moment. Suppose q¿0 for convenience since q = 0 is a trivial case. Then, by the scale and shift property, Q∗ (s) = (1=q)Q(t + qs) is a GFN path in (1). Since the GFN model is stable, we have
Q(s) = 0
Finally, we have
t¿0
Q(t) 2M = w1 (Q(t)):
=
129
u:o:c: as ni → ∞:
Now, for any 3xed t ¿ 0, we have ni Q(t) = lim Q (t) ¿r ni →∞ since Qn (t) ¿ r for su1ciently large n. This contradicts inequality (6). Finally, we can 3nish proving that the S-condition implies the stability of the closed GFN model by repeating steps 2.2 and 2.3 in the proof for part (i) (su1ciency).
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It follows immediately from the part (i) that stability of the closed GFN model implies the L-condition, and it is obvious that the L-condition implies the S-condition by their de3nitions. Hence, the stability, the L-condition and the S-condition of the closed GFN model are equivalent. 3. Applications to some fluid networks A $uid network is a continuous analog of its corresponding queueing network. It can be described by using a system of equations. The importance of the $uid network in studying the stability of the corresponding queueing network lies in the fact that the stability of the $uid network is a su1cient condition for the stability of the corresponding queueing network. In this section, we show that the main theorem in the previous section can be applied to $uid networks under general work-conserving, priority, and FIFO service disciplines. 3.1. A Fluid network under general work-conserving service disciplines The $uid network model consists of J stations, indexed by j = 1; : : : ; J . Each station has an in3nite storage capacity, and stations are interconnected by frictionless pipes to form a network in which several classes of $uids simultaneously circulate. There are K classes of $uids, indexed by k = 1; : : : ; K, and we denote the set of classes by K = {1; : : : ; K}. Class k $uid resides exclusively in station %(k), where %(·) is a many-to-one mapping from the set of classes K to the set of stations J ≡ {1; : : : ; J }. Let C(j) = {k ∈ K: %(k) = j} be the set of classes that reside in station j. Let C = (cjk )J ×K be a J × K constituent matrix, with cjk = 1 if %(k) = j, and cjk = 0 otherwise. To avoid triviality, we assume that C(j) is nonempty for all j = 1; : : : ; J , i.e. all stations must be resided by at least one class of $uids. To describe the $uid network model, we take as primitives two K-dimensional nonnegative vectors Q(0) = (Qk (0)) and & = (&k ), one K-dimensional positive vector ' = ('k ), one K × K substochastic matrix P = (p‘k )K×K with a spectral radius strictly less than one, and one J × K constituent matrix C. For k = 1; : : : ; K, we interpret the kth component Qk (0)
of Q(0) as the initial $uid level of class k (at station %(k)), the kth component &k of & as the exogenous in$ow rate of class k $uid, and the kth component 'k of ' as the potential out$ow rate of class k $uid, or the processing (or service) rate for class k $uid. Namely, if we assume that the time that can be allocated to processing or serving class k $uid during the time interval [s; t] with s 6 t, is t − s, then 'k (t − s) is the maximum possible amount of out$ow of class k $uid in this duration. The (‘; k)th element p‘k of P represents the proportion of the out$ow of Kclass ‘ $uid that turns into class k $uid and 1 − k=1 p‘k is the fraction that leaves the $uid network. Thus, the condition that P has a spectral radius strictly less than one implies that all $uids will visit only a 3nite number of stations before leaving the $uid network. A $uid network model described by the above primitives is referred to as the $uid network (&; '; P; C) with initial $uid level Q(0). Vectors Q(0), & and ' are referred to as the initial >uid level (vector), (exogenous) in>ow rate (vector), and processing rate (vector), respectively. Matrix P is referred to as the >ow transfer matrix. To be consistent with the corresponding terms used for describing a queueing network, sometimes we also call &, ' and P the (exogenous) arrival rate (vector), service rate (vector), and routing matrix, respectively. The processes that conveniently describe the performance of the $uid network are the K-dimensional >uid level process Q = {Q(t); t ¿ 0} and the K-dimensional allocation process T = {T (t); t ¿ 0}. The kth component of T (t), Tk (t), denotes the total amount of time that station %(k) has devoted to processing class k $uid during time interval [0; t]. The dynamics of the work-conserving $uid network can be summarized as follows: Q(t) = Q(0) + &t − (I − P )DT (t) ¿ 0;
(7)
T (·) is nondecreasing with T (0) = 0;
(8)
I (t) = et − CT (t) is nondecreasing; ∞ (CQ(t)) dI (t) = 0;
(9)
0
(10)
where D = diag('). The J -dimensional process I (t) is referred to as the unused capacity process, whose jth component, Ij (t), denotes the (cumulative) unused
H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
capacity of station j during the time interval [0; t] after processing all classes at station j. The relation (7) is the $ow balance relation; its kth coordinate reads as, Qk (t) = Qk (0) + &k t + −'k Tk (t) ¿ 0;
K
Qkn → Uj (u:o:c:), there exists N , for any n¿N , we have k∈C( j)
|Ujn (s) − Uj (s)|¡
on [t − ; t + ];
p‘k '‘ T‘ (t)
and hence
k = 1; : : : ; K:
Ujn (s) ¿ Uj (s) − ¿ ¿0:
‘=1
Eq. (10) is the work-conserving condition. Note that Q ¿ 0 and dI ¿ 0; then Eq. (10) implies that (CQ(t)) dI (t) = 0 and further each term in this summation, Qk (t) dIj (t) = 0; j ∈ J: k∈C( j)
This means that there could be unused capacity at station j only when $uid levels of all classes processed at station j are zero. Readers are referred to Chen [8] for more detailed description of the $uid network with general work-conserving service disciplines. Any pair (Q; T ) that satis3es Eqs. (7) – (10) is said to be a >uid solution of the work-conserving $uid network. Let ,C denote the set of all $uid solutions (Q; T ) of the work-conserving $uid network (7) – (10) and C = {Q: ∃T such that (Q; T ) ∈ ,C }
131
Then Ijn (s) = Ijn (t − )
for all s ∈ [t − ; t + ]:
Letting n → ∞ in the above yields, Ij (s) = Ij (t − )
for all s ∈ [t − ; t + ];
which immediately implies I˙j (t) = 0. Lemma 3.2. Suppose (Q; T ) ∈ ,C is a >uid solution. Then for any given time s ¿ 0 and any given scaler r¿0; (Q(s + ·); T (s + ·) − T (s)) and ((1=r)Q(r·); (1=r)T (r·)) are also in ,C . ˜ Proof. It can be veri3ed directly that both (Q(t); ˆ T˜ (t)) = (Q(s + t); T (s + t) − T (s)) and (Q(t); Tˆ (t)) = ((1=r)Q(rt); (1=r)T (rt)) satisfy the relations (7) – (10). Lemma 3.3. C is a closed GFN model.
be the set of all feasible $uid level processes. The work-conserving $uid network is said to be stable if there is a time ¿ 0 such that Q( + ·) ≡ 0 for any Q ∈ C with Q(0) = 1.
Proof. The Lipschitz property can be seen from conditions (7) – (9). It can also be veri3ed the closeness, scale and shift properties directly from the de3nition of C and Lemmas 3.1 and 3.2.
Lemma 3.1. Suppose (Qn ; T n ) ∈ ,C (n = 1; 2; : : :) is a sequence of >uid solutions; and
Lemma 3.3, combined with Theorem 2.3, leads to the following theorem on a necessary and su1cient condition for the stability of the $uid network under general work-conserving service disciplines.
(Qn ; T n ) → (Q; T );
u:o:c: as n → ∞
with Q(0)¡∞. Then; (Q; T ) ∈ ,C ; i.e.; (Q; T ) is also a >uid solution. Proof. First, it can be checked directly that (Q(t); T (t)) satis3es (7) – (9). To verify (10), it is su1cient to prove that, I˙j (t) = 0 if k∈C( j) Qk (t)¿0. It can be proven by using the continuity of the one dimensional re$ection mapping (see for example Harrison [20]); however, we include a more direct proof here. Suppose Uj (t) ≡ k∈C( j) Qk (t)¿0 for some station index j. Then, there exist ; ¿0, such that Uj (s) ¿ 2 for any s ∈ [t − ; t + ]. Since Ujn ≡
Theorem 3.4. The work-conserving >uid network (7)–(10) is stable if and only if one of the following two conditions holds: (a) the L-condition is satis9ed for C ; (b) the S-condition is satis9ed for C . 3.2. A >uid network under a priority service discipline Relations (7) – (10) that de3ne the K-dimensional $uid level process Q and K-dimensional allocation
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H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
process T essentially contain K + J equations: Kdimensional balance Eq. (7) and J -dimensional work-conserving condition (10). The other (K − J )-dimensional freedom is usually determined by the choice of some speci3c service disciplines. In this section, we specify a priority service discipline. Let 1 be a one-to-one mapping from K onto K. For any given ‘ and k, if 1(‘)¡1(k) and %(‘) = %(k), then class k $uid does not get processed at station %(k) unless the $uid level of class ‘ equals zero. In short, we say that class ‘ has a higher priority than class k. For convenience, the mapping 1 is often expressed as a permutation of K, i.e., which can be written as 1 = (i1 ; : : : ; iK ) if 1(k) = ik , k ∈ K. To describe the $uid network under the priority discipline, we introduce a K-dimensional unused capacity process Y = {Y (t); t ¿ 0}, where Yk (t), the kth component of Y (t), denotes the (cumulative) unused capacity of station %(k) during the time interval [0; t] after serving all classes at station %(k) which have priority no less than class k (including class k). The quantity Yk (t) can also be interpreted as the (cumulative) remaining capacity at station %(k) for processing $uid of classes at station j having a strictly lower priority than class k. Let
remaining capacity (rate) for processing those classes at station %(k) having a strictly lower priority than class k, only when the $uid levels of all classes in Hk (having a priority no less than k) are zero. (Readers are referred to Chen and Zhang [12] for a description of a queueing network under a priority service discipline that provides an explanation for the relations used in the above.) We shall refer to this network as $uid network (&; '; P; C) with initial $uid level Q(0) under priority service discipline 1. For the priority $uid network (11) – (14). A pair (Q; T ) is said to be a >uid solution if they jointly satisfy (11) – (14). Similar to the case of a $uid network under general work-conserving disciplines, let ,P denote the set of all $uid solutions, and P denote the set of all feasible $uid level processes. The priority $uid network is said to be stable if there is a time ¿ 0 such that Q( + ·) ≡ 0 for any Q ∈ P with Q(0) = 1. We state the following lemma, which can be proven in a similar way as Lemmas 3.1–3.3 and hence is omitted.
Hk = {‘: ‘ ∈ C(%(k)); 1(‘) 6 1(k)}
with Q(0)¡∞. Then; (Q; T ) ∈ ,P ; i.e.; (Q; T ) is also a >uid solution. (ii) Suppose (Q; T ) ∈ ,P is a >uid solution. Then for any given time s ¿ 0 and any given scaler r¿0; (Q(s + ·); T (s + ·) − T (s)) and ((1=r)Q(r·); (1=r)T (r·)) are also in ,P . (iii) P is a closed GFN model.
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. Then the dynamics of the network under the priority discipline 1 can be described as follows: Q(t) = Q(0) + &t − (I − P )DT (t) ¿ 0;
(11)
T (·) is nondecreasing with T (0) = 0; Yk (t) = t − T‘ (t) is nondecreasing;
(12)
‘∈Hk
0
∞
Qk (t) dYk (t) = 0;
k ∈ K; (13)
k ∈ K:
(14)
Relations (11) – (14) subsume relations (7) – (10), if we note that Yk (t) = I%(k) (t), when k has the lowest priority at station %(k). In particular, relation (14) speci3es both the work-conserving condition and the priority discipline; in words, for each k, relation (14) means that at any time t, there could be some positive
Lemma 3.5. (i) Suppose (Qn ; T n ) ∈ ,P (n = 1; 2 : : :) is a sequence of >uid solutions; and (Qn ; T n ) → (Q; T );
u:o:c: as n → ∞
Lemma 3.5, combined with Theorem 2.3, implies the following theorem on a necessary and su1cient condition for the stability of a $uid network under a priority service discipline. Theorem 3.6. The priority >uid network (11)–(14) is stable if and only if one of the following two conditions holds: (a) the L-condition is satis9ed for P ; (b) the S-condition is satis9ed for P . 3.3. A >uid network under FIFO service discipline The settings of the FIFO $uid network parallel those of a priority $uid network, but with a diKerent set of
H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
dynamic relations to complement the dynamic relations of a $uid network under general work-conserving disciplines. The parameters (&; '; P; C), the $uid level process Q and the allocation process T are also de3ned in the same way as those for the work-conserving $uid network. The dynamics of the FIFO $uid network can be summarized as follows: (7) – (10), and D(t) = M −1 T (t);
A(t) = &t + P D(t); W (t) = CMQ(t);
(15) t ¿ 0; t ¿ 0;
(16) (17)
Dk (t + Wj (t)) = Qk (0) + Ak (t); t ¿ 0; j = %(k); k = 1; : : : ; K;
(18)
where M = diag(m). The K-dimensional processes A={A(t); t ¿ 0} and D={D(t); t ¿ 0} are referred to as the in>ow (or arrival) process and the out>ow (or departure) process, respectively. The J -dimensional process W (t) is referred to as the workload process. The jth component of W (t), Wj (t), is the amount of time required for station (server) j to process all the $uid presented at station j at time t. Relation (18) is the key relation that speci3es the FIFO service discipline. Readers are referred to Harrison [21] for a more detailed description, and Chen and Zhang [11] for an alternative description. Again, for the FIFO $uid network (7) – (10) and (15) – (18), we also de3ne the $uid solution (Q; T ), the set of $uid solutions ,F , the set of all feasible $uid level processes F , and the stability concept, in the same way as we did for the work-conserving $uid network. Similar to Lemma 3.5, we have the following lemma. Lemma 3.7. (i) Suppose (Qn ; T n ) ∈ ,F (n = 1; 2 : : :) is a sequence of >uid solutions of the >uid network; and (Qn ; T n ) → (Q; T );
u:o:c: as n → ∞:
Then; (Q; T ) ∈ ,F ; i.e.; (Q; T ) is also a >uid solution. (ii) Suppose (Q; T ) ∈ ,F is a >uid solution. Then for any given time s ¿ 0 and any given scaler r¿0; (Q(s + ·); T (s + ·) − T (s)) and ((1=r)Q(r·); (1=r)T (r·)) are also in ,F . (iii) F is a closed GFN model.
133
Proof. Parts (ii) and (iii) can be proved in the same manner as Lemmas 3.2 and 3.3, respectively. For part (i), it is su1cient to verify that (Q; T ) satis3es Eqs. (15) – (18). Verifying (15) – (17) is trivial. It is clear that lim Dkn (t + Wjn (t)) = Dk (t + Wj (t))
n→∞
for all t ¿ 0;
since all the functions involved are (Lipschitz) continuous. (Here we append a superscript n to all the processes corresponding to (Qn ; T n ).) It follows immediately from the above convergence that Eq. (18) also holds for (Q; T ). Lemma 3.7, combined with Theorem 2.3, implies the following theorem that describes a necessary and su1cient condition for the stability of the $uid network under FIFO service discipline. Theorem 3.8. The FIFO >uid network (7)–(10) and (15)–(18) is stable if and only if one of the following two conditions holds: (a) the L-condition is satis9ed for F ; (b) the S-condition is satis9ed for F . 4. Application to fluid limit models of queuing networks In this section, we apply the main result to some $uid limit models, the stability of which also implies the stability of the corresponding queueing networks. Both the $uid network and the $uid limit model are the continuous analog of the corresponding queueing network. The diKerence between them lies in the fact that, the $uid network is described by a set of equations (or conditions), while the $uid limit model arises as the limit of sequences of scaled queueing networks. In general, a $uid limit path (to be de3ned later) in the $uid limit model is also a solution of the corresponding $uid network, but the converse may not be true. Therefore, the stability condition of a $uid limit model may be strictly weaker than the stability condition of a corresponding $uid network. The study of the former can be important in view of the recent counterexamples brought forth by Bramson [7] and Dai et al. [15], where the queueing networks are found stable under some arrival=service time distributions, but the
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corresponding $uid networks are not stable. We conjecture that for these queueing networks, their corresponding $uid limit models are stable. 4.1. Queuing network The queueing network consists of J stations indexed by j = 1; : : : ; J; and K job classes indexed by k = 1; : : : ; K. Let uk = {unk ; n ¿ 1} be the sequence of interarrival time process, and v k = {vnk ; n ¿ 1} be the sequence of service time process, where unk indicates the interarrival time between the (n − 1)st and the nth jobs of class k, and vnk indicates the service time for the nth job of class k, k = 1; : : : ; K. A class k job is served at station %(k), and after its service completion, it may become a class ‘ job with probability K pk‘ and leave the network with probability 1 − ‘=1 pk‘ . Let P = (pk‘ ). Let C = (cjk ) be a J × K matrix whose (j; k)th component cjk =1 if j=%(k) and =0 otherwise. While each station may serve more than one class of jobs, each job is served at one speci3c station (determined by the many-to-one mapping %(·)). To describe the state space of the queueing network precisely, we need to specify the service discipline such as the priority service discipline, and FIFO service discipline. We assume all random variables are de3ned on a probability space (7; F; P) with expectation operator E. Assume that u1 ; : : : ; uK ; v 1 ; : : : ; v K are mutually independent iid sequences;
(19)
mk := Ev1k ¡∞
(20)
for k = 1; : : : ; K:
To avoid triviality, assume E := {k: Eu1k ¡∞} = ∅. Furthermore, assume that for each k ∈ E, there exists some integer nk ¿0 and some function pk (x) ¿ 0 on ∞ [0; ∞) with 0 pk (x) d x¿0, such that P{u1k ¿ x}¿0 for each x¿0;
b nk k u‘ 6 b ¿ pk (x) d x P a6 ‘=1
for each 0 6 a¡b:
(21)
a
(22)
Let & = (&k ) be the exogenous arrival rate (vector) where &k = 1=Eu1k for k ∈ E and &k = 0 for k ∈ E; k = 1; : : : ; K.
With the above primitives, we can de3ne a Markov process X = {X (t); t ¿ 0} to describe the dynamics of the queueing network. Then, a queueing network described above is said to be stable if its underlying Markov process X is positive Harris recurrent. (Readers are referred to Dai [13] for a formal description of the state process X (t), and to Dai [13] and Meyn and Tweedie [25] for a formal de3nition of positive Harris recurrence.) 4.2. Fluid limit model Now, we consider a sequence of queueing networks described in the last subsection. Let Qn : R+ → ZK+ ; n = 1; 2; : : : ; be a sequence of corresponding queue length processes with initial queue length Qn (0), where Z+ is the set of all nonnegative integers. The kth coordinate of Qn (t), Qkn (t), indicates the number of class k jobs in the nth network at time t. For almost all sample paths ! ∈ 7 , any increasing positive real number sequence {rn } with rn → ∞, and any sequence of initial queue lengths Qn (0) with (1=rn )Qn (0) → Q(0), there is a subsequence {rnj } of {rn } such that 1 rnj T Q (rnj t) → Q(t); rn j
u:o:c: as nj → ∞:
T Any limit Q(t) de3ned above is called the ( feasible) T >uid limit path with initial $uid level Q(0) The set of all these $uid limit paths, denoted by L , is said to be the >uid limit model. It is known that under the assumptions (21) – (22), the queueing network is stable if the $uid limit model T is stable (i.e., there exists a ¿ 0 such that Q(+·) ≡ T T 0 for any Q ∈ L with Q(0) = 1) (Theorem 4.2 of Dai [13]). This result indicates that the study of the stability of $uid limit model is important for the study of the stability of the corresponding queueing network. For the $uid limit model L , we have the following property. Proposition 4.1. The >uid limit model L is a GFN model. Proof. We need to verify conditions (a) – (c) in the de3nition of GFN model for L . Firstly, condition (a) follows from Theorem 4:1 in Dai [13].
H.Q. Ye, H. Chen / Operations Research Letters 28 (2001) 125–136
To verify the scale and shift property, we suppose that Q ∈ L . Then there is ! ∈ 7, an increasing positive real number sequence {rn } with rn → ∞, and a sequence of scaled queue length processes Qn (t) such that 1 n T Q (rn t) → Q(t); rn
u:o:c: on t ∈ [0; ∞):
Let rn = r · rn . Then, we have 1 n 1 1 n 1 Q (rn t) = Q (rn (rt)) → Q(rt); rn r rn r u:o:c: on t ∈ [0; ∞); and this shows that 1r Q(rt) ∈ L . For any 3xed number ¿ 0; 1 n T + t); Q (rn ( + t)) → Q( rn u:o:c: on t ∈ [0; ∞); T + t) ∈ L . and hence Q( However, we could not establish the closeness property for the $uid limit model L . Applying Theorem 2.3 to L yields the following theorem. Theorem 4.2. The >uid limit model L stable if and only if the L-condition holds for L . 5. Application to a linear Skorohod problem In this section, we apply our main result to a linear Skorohod problem (LSP), and obtain a necessary and su1cient condition for the stability of this LSP. The LSP is closely related to the theory of queueing networks. Its stability implies the stability of the corresponding semi-martingale re$ected Brownian motion (SRBM) (see Dupuis and Williams [19]), which arises as the diKusion limit of a queueing network. Readers are referred to Chen [9] and the references there for details on LSP and SRBM. Let R be a J × J matrix and < ∈ RJ . Given X (0) ∈ J R+ , the J -dimensional continuous functions from R+ to RJ , Y and Z, are said to solve the LSP(<; R) with initial state X (0), if they jointly satisfy
X (t) dY (t) = 0:
135
(25)
Let ,S be the set of all solutions (X; Y ) of LSP(<; R), and let S be the set, {X : ∃Y s:t: (X; Y ) ∈ ,S }: A LSP(<; R) is said to be stable if, for any number ¿0 and any X ∈ S with X (0) = 1, there exists a ¿ 0 such that X ( + ·)¡ . To be consistent with the existing de3nition in the literature, the stability of the LSP is de3ned slightly diKerent from that of the $uid networks (requiring X ( + ·)¡ instead of X ( + ·) = 0), but we shall see momentarily that they are actually equivalent (under some conditions). 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 next lemma follows from an oscillation inequality by Bernard and El Kharroubi [1]. Lemma 5.1. Suppose R is completely-S. Then; there is a constant M such that any solution (X; Y ) of the LSP(<; R) is Lipschitz continuous with Lipschitz constant M . Then, with some straightforward arguments, we can establish that S is a closed GFN model if R is completely-S. Applying Theorem 2.3 to this case yields the following theorem. Theorem 5.2. Suppose R is completely-S. Then; the LSP(<; R) is stable if and only if one of the following three conditions holds: (a) there exists ¿ 0 such that X ( + ·) ≡ 0; (b) the L-condition is satis9ed for S ; (c) the S-condition is satis9ed for S . Proof. Since S is a closed GFN model under the condition that R is completely-S, we have, by part (ii) of Theorem 2.3, (a) ⇔ (b) ⇔ (c):
X (t) = X (0) +
(23)
By de3nition of stability of LSP, it is also obvious that
dY (t) ¿ 0;
(24)
(a) ⇒ the LSP(<; R) is stable ⇒ (c):
Y (0) = 0;
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6. Concluding remarks In this paper, we have obtained necessary and su1cient conditions for the stability of various $uid models, $uid limit models and a LSP. This shows that the Lyapunov method would play the same role in the stability of $uid models as it plays in the stability of deterministic dynamic systems. In particular, the “only if” part (the converse theorem) supports the previous eKorts on the use of Lyapunov functions to characterize the stability of $uid networks. References [1] A. Bernard, A. El Kharroubi, RWegulation de processus dans le premier orthant de Rn , Stochastics Stochastics Reports 34 (1991) 149–167. [2] D. Bertsimas, D. Gamarnik, J.N. Tsitsiklis, Stability conditions for multiclass $uid queueing networks, IEEE Trans. Automat. Control 41 (1996) 1618–1631. [3] D.D Botvich, A.A. Zamyatin, Ergodicity of conservative communication networks, Rapport de recherche 1772, INTRA, October, 1992. [4] M. Bramson, Instability of FIFO queueing networks, Ann. Appl. Probab. 4 (1994) 414–431. [5] M. Bramson, Convergence to equilibria for $uid models of FIFO queueing networks, Queueing Systems: Theory Appl. 22 (1996) 5–45. [6] M. Bramson, Convergence to equilibria for $uid models of head-of-the-line proportional processor sharing queueing networks, Queueing Systems: Theory Appl. 23 (1997) 1–26. [7] M. Bramson, A stable queueing network with unstable $uid model, Ann. Appl. Probab. 9 (1999) 818–853. [8] H. Chen, Fluid approximations and stability of multiclass queueing networks: Work-conserving discipline, Ann. Appl. Probab. 5 (1995) 637–655. [9] H. Chen, A su1cient condition for the positive recurrence of a semimartingale re$ecting Brownian motion in an orthant, Ann. Appl. Probab. 6 (1996). [10] H. Chen, H.Q. Ye, Piecewise linear lyapunov function for the stability of priority multiclass queueing networks, Proceedings of the 38th IEEE Conference on Decision and Control, 1999, pp. 931–936. [11] H. Chen, H. Zhang, Stability of multiclass queueing networks under FIFO service disciplines, Math. Oper. Res. 22 (3) (1997) 691–725. [12] H. Chen, H. Zhang, Stability of multiclass queueing networks under priority service disciplines, Oper. Res. 48 (2000). [13] J.G. Dai, On positive Harris recurrence of multiclass queueing networks: a uni3ed approach via $uid models, Ann. Appl. Probab. 5 (1995) 49–77.
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