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

Here we shall mention only the results referring to stability. The definitions of the various quantities T n , S n , S N n , 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 { S N n , T n } arising in connexion with queues in series will be called the component queues, and the queue { S n , s T n } 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 { S n , T n } critical if E(S 0 -T0) = 0. We shall now call it subcritical if E(S0 − T o ) > 0 and supercritical if E(S 0 − T 0 ) S n , T n } will be said to be of type M if it has the property considered in Corollary 1 to Theorem 5: the sequences { S n } and { T n } 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.