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# The stability of a queue with non-independent inter-arrival and service times

## Introduction

Here we shall mention only the results referring to stability. The definitions of the various quantities Tn, Sn, SNn, and the basic hypotheses made concerning their structure will be found in §§ 2·1, 3·1 or 4·1. For convenience we shall introduce some further terminology in this section. The single-server queues {SNn, Tn} arising in connexion with queues in series will be called the component queues, and the queue {Sn, sTn} implicit in the discussion of many-server queues will be called the consolidated queue. We have already in § 2.33 called the single-server queue {Sn, Tn} critical if E(S0-T0) = 0. We shall now call it subcritical if E(S0 − To) > 0 and supercritical if E(S0 − T0) < 0. A system of queues in series is subcritical if each component queue is subcritical, critical if (at least) one component queue is critical and the rest are subcritical, and supercritical if (at least) one component queue is supercritical. A many-server queue will be described in these terms according to the character of its consolidated queue. Finally, a single-server queue {Sn, Tn} will be said to be of type M if it has the property considered in Corollary 1 to Theorem 5: the sequences {Sn} and {Tn} are independent of each other, and one is composed of mutually independent non-constant random variables.

Single-server queues:

(i) Subcritical: stable (Theorem 3).

(ii) Supercritical: unstable (Theorem 2).

(iii) Critical: stable, properly substable, or unstable (examples in §2·33, including one due to Lindley); unstable if type M (Theorem 5, Corollary 1).

Queues in series:

(i) Subcritical: stable (Theorem 7).

(ii) Supercritical: unstable (Theorem 7).

(iii) Critical: stable, properly substable, or unstable, if the component queues are substable (examples in § 3·2); unstable if any component queue is unstable (Theorem 7), and in particular if any critical component queue is of type M (Theorem 7, Corollary).

Many-server queues:

(i) Subcritical: stable or properly substable (Theorem 8, and example in § 4·3).

(ii) Supercritical: unstable (Theorem 8).

(iii) Critical: stable, properly substable, or unstable, if consolidated queue is substable (examples in § 4·3); unstable if consolidated queue unstable (Theorem 8), and in particular if this is of type M (Theorem 8, Corollary).

From Lemma 1 it follows that none of these queues can be properly substable if all the servers are initially unoccupied.

## References

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(1)Birkhoff, G., Lattice theory (New York, 1948).
(2)Doob, J. L., Stochastic processes (New York, 1953).
(3)Kiefer, J., and Wolfowitz, J., Trans. American Math. Soc. 78 (1955), 1.
(4)Lindley, D. V., Proc. Cambridge Philos. Soc. 48 (1952), 277.
(5)Loève, M., Probability theory (2nd ed.) (New York, 1960).
(6)Sacks, J., Ann. Math. Statist. 31 (1960), 579.

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