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- Title
Revisiting queueing output processes: a point process viewpoint.
- Authors
Daley, D. J.
- Abstract
After some historical notes concerning queueing output processes N, the paper discusses methods for establishing asymptotic linear relations for var N(0, t], whether in the crude form B t or the more detailed form B t+ B+ o(1) for t→∞. The crude form holds whenever the process N of customers admitted to service has a linear asymptote, and then (var N(0, t])/ t and (var N(0, t])/ t share a common limit (that may be infinite) in stationary G/G/ k/ K systems. A standard integral formula for the variance of a stationary orderly point process shows that, if N is a renewal process whose generic lifetime X has finite second moment, then B=(var X)/([ E( X)]), and the more detailed linear asymptote holds when E( X) is finite. Geometric ergodicity of the queue size process Q(⋅) in stationary M/M/ k/ K systems establishes that the more detailed linear asymptote is true for them. It is conjectured that var N(0, t]∼ B t for any stationary point process N possessing an embedded regenerative structure.
- Subjects
QUEUING theory; PRODUCTION control; ASYMPTOTIC theory in linear differential equations; CUSTOMER services; RENEWAL theory; POISSON integral formula; EMBEDDED computer systems
- Publication
Queueing Systems, 2011, Vol 68, Issue 3/4, p395
- ISSN
0257-0130
- Publication type
Article
- DOI
10.1007/s11134-011-9232-3