Large Bursts Don't Cause Instability
By Bruce Hajek
University of Illinois at Urbana-Champaign
Abstract: It is shown that stability of networks with uid trac implies stability of
networks with deterministically constrained trac.
This paper was presented at the 1995 IEEE International Symposium on Information Theory
This research was supported by the National Science Foundation under contract NSF NCR 93
14253.
Author's address:
Coordinated Science Laboratory
1308 West Main Street
Urbana, IL 61801, USA
Tel: (217) 333 3605
e-mail: b-hajek@uiuc.edu
1 Introduction
Fluid models of queuing networks are among the simplest models to analyze, owing to the fact
that calculus can be applied. At the same time, wider classes of network models are more exible
for modeling real trac. It is thus useful to reduce questions about the more realistic models to
questions about related uid models. Such a reduction was achieved by Dai [4], who showed that
stability of a uid model implies stability (in the sense of Harris recurrence) of related multiclass
networks with random service and interarrival processes of renewal type. The purpose of this note
is to similarly reduce the question of stability for networks with input trac satisfying deterministic
constraints in the sense of Cruz [2] to a question of stability for a uid model.
2 The Network Model
The network we consider has d single server stations and K classes of trac. Class l trac is served
at a unique station s(l). Let C be the d K matrix such that Ci;l = 1 if s(l) = i and Ci;l = 0
otherwise. Upon completion of service at s(l) the trac of class l either becomes trac of class l0
for some other class l0, in which case we write l ! l0, or it immediately exits the network. Let P
denote the K K matrix such that pl;l = 1 if l ! l0 and pl;l = 0 otherwise. A simple network
0
0
with three stations and eight classes is shown in Figure 1 with 1 ! 2 ! 3, 4 ! 5 ! 6 and 7 ! 8
and s = (1; 2; 3; 2; 3; 1; 3; 2). It is assumed that the network is open, so that P K is the zero matrix.
Exogenous trac can enter the network as any class, though for the example given it might
make sense for the exogenous arrival functions to be nonzero only for classes 1, 4 and 7. Let El(t)
denote the amount of exogenous class l trac to enter the network during [0; t]. Suppose that
2
El(0) = 0 and that El is nondecreasing and right-continuous. Trac of class l can be served (at
station s(k)) at a maximum rate l = 1=ml where ml > 0. Let M = diag (m1; : : :; mK ) and let e
denote a column vector of all ones (with dimension depending on the context). The ow of trac
in the network is assumed to satisfy the following equations and conditions:
Q(t) = q + E (t) + (P T ? I )M ?1T (t)
(2.1)
I (t) = et ? CT (t)
(2.2)
Q(t) 0 for t 0
Z1
0
(CQ(t) ^ e) dI (t) = 0
(2.3)
(2.4)
T (0) = 0; T is right-continuous and nondecreasing
(2.5)
I (0) = 0; I is right-continuous and nondecreasing.
(2.6)
The the following interpretations hold:
Ql(t) is the amount of class l trac in the network at time t, and Ql(0) = q0 .
Tl(t) is the amount of work (where work is measured in units of time) done on class l trac during
[0; t]. A unit increment of Tl corresponds to l units of class l trac served at station s(l),
which becomes class l0 trac if l ! l0.
(CT (t))i, where CT (t) is simply the matrix C times the vector T (t), is the amount of work (where
work is measured in units of time) done at station i during [0; t].
Ii(t) is the amount of idleness (measured in units of time) of the server at station i accumulated
during [0; t]
3
(CQ(t))i is the amount of trac at station i at time t.
The exogenous trac E is said to satisfy deterministic constraints with parameters = (l)
and = (l), abbreviated to \E is DC(; ) trac", if
0 El (t) ? El (s) l (t ? s) + l 0 t s < 1:
(2.7)
This work was completed independently of the work of Gamarnik [7], which is remarkably
similar. There are some dierences, aside from dierences in terminology. The arrival constraints
imposed by Gamarik are more general, in that they are imposed on sums of path arrival processes,
one constraint for each link in the network, whereas we assume each individual arrival stream is
constrained. On the other hand, the type of constraint we impose is somewhat more general than
that of Gamarik, in that the Cruz type contraints used here are more general than constraining
the number of arrivals in xed length windows. Also, the service times for packets on a given path
can be dierent at dierent stations.
3 Sucient Conditions for Stability
Denition 3.1 The network (C; P; m) is totally stable for DC(; ) trac if there is a nite
constant ? so that whenever (E; q; Q; T; I ) satisfy (2:1) ? (2:7), then lim supt!1 jQ(t)j ?, where
jQ(t)j = Pl jQl(t)j.
We use the word \totally" in the above denition because the constraints do not indicate an
order of service at the stations.
4
A subclass of DC(; ) trac is DC(; 0) trac, obtained by setting the vector = 0. DC(; 0)
trac can be considered to be uid trac with arrival rates bounded by . The special case
E (t) = t is the uid arrival function with arrival rate vector equal to . For uid arrivals, we
adopt a dierent denition of stability, though it is not dicult to show that it is equivalent to the
one above.
Denition 3.2 The network (C; P; m) is totally stable for trac with arrival rates bounded by
(resp. exactly equal to ) if there is a nite constant f so that whenever (E; q; Q; T; I ) satises
(2:1) ? (2:6), jq j 1, and E is DC(,0) trac (resp. and E (t) = t), then Q(t) = 0 for t tf .
Two propositions are given below. The rst reduces the question of stability for DC(; ) trac
to a question of stability for uid trac with arrival rate vector bounded by , and the second
proposition further reduces it to a question of the stability for uid trac with arrival rate vector
equal to . The proofs are found in the appendix. Proposition 3.2, basically a \monotonicity"
result, is a special case of Proposition 3.6 in [3]. Not all details are provided in the proof here.
Proposition 3.1 If the network (C; P; m) is totally stable for DC (; 0) trac, then it is totally
stable for DC (; ) trac, for any vector .
Proposition 3.2 If the network (C; P; m) is totally stable for uid trac of rate , then it is
totally stable for DC (; 0) trac.
5
4 Conclusions
This work was originally motivated by the juxtaposition of two interesting results: an ingenious but
complex proof of Tassiulas and Georgiades [8] of total stability for deterministically constrained
trac on a unidirectional ring network on the one hand, and an ingenious and simple proof later
obtained by Dai and Weiss [5] that the same network is stable for uid trac on the other hand.
A unidirectional ring network is characterized by the constraint that l ! l0 only if l0 = l + 1
mod K . Propositions 3.1 and 3.2 combined with the stability of ring networks with uid trac
whenever = M (I ? P )?1 < e yields another proof of part of the result in [8]. The result in [8]
is stronger, however, in that it provides a simple explicit bound on the lim sup of the maximum
queue length. (It also allows for non-zero transmission delays, but it assumes that processor speeds
are all one). The proof of stability for uid trac in [5] for the ring network oers a clean bound
on the rate of evacuation. It would be desirable if Proposition 3.1 could be strengthened to allow
us to translate bounds for evacuation rates in uid models to bounds on asymptotic queue lengths
with deterministically constrained trac.
Propositions 3.1 and 3.2 can apparently be generalized to cover networks with specied classes
of service orders, which are represented by additional constraints. For example, it is known that
uid networks with FIFO service order are stable if < e and mk = ml whenever s(k) = s(l) [1]. If
this conjecture is proved, an extension of Proposition 3.1 could show that such networks are stable
for deterministically constrained trac as well. In order to extend Proposition 3.1 to hold when
the networks are restricted to a class of networks using a specied service order, the constraint on
order of service should be inherited under u.o.c.convergence of (E; q; Q; T; I ) and other functions
6
that may be needed to model the network with specic order of service constraints.
5 Appendix
The two propositions are proved in this appendix. A key to the proof of Proposition 3.1 is that if
E is DC(; ) trac, then for any > 0 the rescaled function t ! E (t)= is DC(; = ) trac.
Moreover, the conditions (2.1)-(2.6) are invariant under a similar rescaling of (E; Q; T; I ). The
proposition is proved after the following lemma is proved.
Lemma 5.1 Let f 2 DR+ [0; 1) such that for some nonnegative constants ; ; a; b; c, with < 1:
f (t + f (t)) f (t) if f (t) > a
(5.1)
f (t + s) ? f (s) b + ct if s; t 0
(5.2)
Then f (s) (a + b)(1 + c=(1 ? )) for s f (0) 1? .
Proof. If f (t) > a, then by (5.1), f (t + f (t)) f (t) ? 1? (f (t)). Graphically, if f (t) > a then
(t + f (t); f (t + f (t))) lies below the line through (t; f (t)) with slope ? 1? . Therefore, dening the
sequence of points (tn ; f (tn )) by t0 = t and tn+1 = tn + f (tn ), and letting N = minfn 0 : f (tn ) ag, the points of the sequence t0 ; : : :; tN all lie below the line through (t; f (t)) with slope ? 1? .
Therefore f (t) ? (tN ? t) 1? f (tN ) 0 so that tN t + f (t) 1? . Thus, if f (t) > a, then f (s) a
for some s 2 [t; t + f (t) 1? ]. In particular, the time t dened by t = minfs 0 : f (s) ag is at
most f (0) 1? .
To nish the proof, a bound on f restricted to the interval [t; 1) needs to be found. Let
a0 > a. If f (t?) a0 f (t), then by condition (5.2), f (t) f (t ? ) + b + c for any > 0 so
7
that f (t) a0 + b. Thus, by the rst paragraph of the proof, f (s) a for some s in the interval
[t; t + (a0 + b) 1? ]. By (5.2), throughout this interval, f (s) a0 + b + c(a0 + b) 1? . However,
the whole set fs t : f (s) a0g is covered by such intervals. Conclude that for s t ,
f (s) (a0 + b)(1 + c=(1 ? )). Since a0 > a is arbitrary, the proof of the lemma is complete. 2
Proof of the Proposition 3.1. Suppose (C; P; m) is not totally stable for DC(; ) trac.
Let b = j + (I + P T )j, c = j j and x > 0 and with 0 < < 1. By Lemma 5.1 and
the assumption that (C; P; m) is not totally stable for DC(; ) trac, given a > 0 there exists
a solution S a = (E a; Qa; q a; T a; I a) to (2.1)-(2.7) such that f a dened by f a (t) = jQa (t)j violates
either (5.1) or (5.2). Examination of the constraints (2.1),(2.2), (2.5) and (2.6) satised by S a shows
that f a satises (5.2), so it must violate (5.1), meaning f a (ta + f a (ta )) > f a (ta ) and f (ta ) > a
for some ta 0. By shifting the functions of S a to the left if necessary, it can be assumed without
loss of generality that ta = 0 for all a. Hence, S a = (E a; Qa; q a; T a; I a) satises (2.1)-(2.7) and the
additional constraints jQa ( jq aj)j > jq a j and jq aj a.
Next a space-time rescaling is used. Let E a (t) = E a(tjq aj)=jq aj, and dene Qa; T a , and I a,
similarly. Also set q a = qa =jqaj. Note that jQa(0)j = jq a j = 1 and jQa( )j for all a. The
scaled versions (Ea ; qa ; Qa ; T a ; I a ) again satisfy (2.1)-(2.6), while (2.7) is satised with replaced
by =jqaj, which is at most =a. The family of functions (E a )a0 is thus asymptotically uniformly
equicontinuous as a ! 1, and jE a(t)j jjt + j j for all a 1, so the family is also asymptotically
uniformly bounded on compact intervals. The functions jT a j are Lipschitz continuous uniformly
in a with common value 0 at the origin. Thus, there are continuous limit functions such that,
along a subsequence of a ! 1 the following limits hold: q a ! q , E a ) E uniformly on compact
intervals (u.o.c.) and T a ) T u.o.c. Note that jq j = 1. With Q and I dened in terms of E and T
8
by (2.1) and (2.2), it also follows that (Qa ; I a ) ! (Q; I ) u.o.c. along the same subsequence. The
limit (E; q; Q; T; I ) satises the conditions (2.1)-(2.6). (To verify condition (2.4) use the fact that if
((zn ; yn )) is a sequence in DR [0; 1) DR[0; 1), yn in nondecreasing for each n, and (zn ; yn ) ! (z; y )
R
R
u.o.c. then for any bounded continuous function h, 0t h(zn (s))dyn (s) ) 0t f (z (s)dy (s) u.o.c. [6,
Lemma 2.4].) The fact that E a is DC(; =a) trac for each a yields in the limit that E is DC(,0)
trac. Finally, since jQa( )j for all a, the limit satises Q( ) .
To summarize, it has been shown that if (C; P; m) is not totally stable for DC(; ) trac, then
for any > 0 and any with 0 < < 1 there is a solution (E; q; Q; T; I ) to (2.1)-(2.6) such that
E is DC(; 0) trac, jQ(0)j = jqj = 1, and jQ( )j . Therefore (C; P; m) is not totally stable for
DC(,0) trac. The contrapositive of the proposition, and hence the proposition itself, is proved.
2
Proof of Proposition 3.2. It suces to show that given S = (E; q; Q; T; I ) satisfying (2.1)~ q; Q;
~ T;
~ I~), can be found with E~ (t) = t such
(2.6) where E is DC(; 0), another solution S~ = (E;
that Q(t) Q~ (t) for all t. Intuitively, since E (t) ? E (s) E~ (t) ? E~ (s) whenever s < t, the trac
for S~ can be viewed as the sum of two trac types: original trac and \ghost" trac. The servers
simply give priority to original trac. The solution S~ is easy to construct by induction if the
functions comprising S are piecewise linear, and in the general case a sequence of approximating
solutions is used. The details are similar to those in the proof of Proposition 3.1, and are omitted.
2.
9
References
[1] M. Bramson, \Convergence to equilibria for uid models of FIFO queuing networks," Queueing
Systems and Their Applications, vol 22, p. 5-45, 1996.
[2] R. L. Cruz, \A calculus for network delay, Part I: network elements in isolation," IEEE
Transactions on Information Theory, vol. 37, no. 1, pp. 114{131, January 1991.
[3] H. Chen, \Fluid approximations and stability of multiclass queueing networks: work conserving
disciplines," Annals of Applied Probability, vol. 5, pp. 637-665, 1995.
[4] J.G. Dai. \On the positive Harris recurrence for multiclass queueing networks: A unied
approach via uid limit models," Annals of Applied Probability, vol. 5, pp. 49-77, 1995.
[5] J.G. Dai and G. Weiss, \Stability and instability of uid models for re-entrant lines," Mathematics of Operations Research, vol. 21, pp. 115-134, 1996.
[6] J.G. Dai and R.J. Williams, \Existence and uniqueness of semimartingale reecting Brownian
motion in convex polyhedrons," Theory of Probability and its Applications, vol. 50, pp.3-53,
1995.
[7] D. Gamarnik, \Stability of adversarial queues via uid models," This issue.
[8] L. Tassiulas and L. Georgiadis. \Any work-conserving policy stabilizes the ring with spatial
reuse," IEEE/ACM Transactions on Networking, vol. 4, no. 2, pp. 205-208, 1996.
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6
Station 2
b
b
b
Station 1
BB
Q
Q
Station 3
Figure 1: Sample network.
11
S
S