CLASSICAL k-SERVER QUEUES REVISITED: SOME PROBLEMS

The talk surveys both recent and ‘classical’ studies of k-server queueing systems. There will not be time to do justice to more than a selection of the following topics. 1. EXISTENCE. Kiefer and Wolfowitz’ (1955) proof of the existence of a stationary regime for the GI/GI/k FCFS system relies on ‘An Essential Lemma’. In reality it hinges on a strong law of large numbers, of which variants are useful in a range of problems including k-fast servers and the tail behaviour of the work-load vector. It depends on ergodicity and not the independence assumptions of GI/GI/k (cf. Loynes’ Lemma and Construction). 2. CYCLES. Much work on busy period analysis has relied on either regeneration points or weakened versions of them. The prime underlying result again is one of ergodicity and stationarity, and exploits ideas of point processes, albeit subsequent to Khinchin’s (1955) monograph. 3. MOMENTS. The fact that a sub-minimally stable GI/GI/k system has finite mean delay when E(S) < ∞ for a generic service-time r.v. S is now better understood. An intermediate result concerns the moment index κ(X) = sup{k : E(X) < ∞} of independent r.v.s X and Y , with κ(min(X, Y )) ≥ κ(X)+κ(Y ). Equality holds if X has a d.f. with a regularly varying tail, but the class is a little larger than this. 4. EXPONENTIAL SYSTEMS. Let two identical ./M/1 systems be fed by the same Poisson arrival process: what is their joint stationary distribution, and correlation of the queue sizes? Alan Brown has conjectured the structure of the d.f. sufficient for use with an expression for the stationary bivariate p.g.f.: is there any ‘routine’ approach to circumvent the relative intractability of such simple systems? 5. HEAVY TAILS. The same dichotomy between minimally and sub-minimally stable systems as occurs for moment behaviour affects the asymptotic tail-behaviour of the work-load vector when the service-time d.f. is heavy-tailed. 6. SECOND-ORDER PROPERTIES. The stationary waiting-time sequence {Wn} in a singleserver system has monotonic decreasing correlations (i.e. corr(W0,Wn) ↓ for 0 < n ↑), and its Hurst index is a simple function of the moment index of service times. Does monotonicity hold in GI/GI/k ? And what is the Hurst index of corr(W0,Wn): is it affected by the system being minimally stable or not?