Logarithmic Asymptotics for Steady-State Tail Probabilities in a Single- Server Queue

1993 ◽  
Author(s):  
Peter W. Glynn ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Single Server ◽  
Single Server Queue ◽  
Tail Probabilities ◽  
Logarithmic Asymptotics ◽  
Server Queue

Logarithmic asymptotics for steady-state tail probabilities in a single-server queue

1994 ◽  
Vol 31 (A) ◽  
pp. 131-156 ◽  
Author(s):  
Peter W. Glynn ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Decay Rate ◽  
Partial Sums ◽  
Asymptotic Result ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queue ◽  
Asymptotic Decay ◽  
Server Queue ◽  
Service Times

We consider the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steady-state waiting-time distribution to have asymptotics of the form x–1 log P(W> x) → –θ ∗as x → ∞for θ ∗ > 0. We require only stationarity of the basic sequence of service times minus interarrival times and a Gärtner–Ellis condition for the cumulant generating function of the associated partial sums, i.e. n–1 log E exp (θSn) → ψ (θ) as n → ∞, plus regularity conditions on the decay rate function ψ. The asymptotic decay rate θ is the root of the equation ψ (θ) = 0. This result in turn implies a corresponding asymptotic result for the steady-state workload in a queue with general non-decreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates.

Logarithmic asymptotics for steady-state tail probabilities in a single-server queue

1994 ◽  
Vol 31 (A) ◽  
pp. 131-156 ◽  
Author(s):  
Peter W. Glynn ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Decay Rate ◽  
Partial Sums ◽  
Asymptotic Result ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queue ◽  
Asymptotic Decay ◽  
Server Queue ◽  
Service Times

We consider the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steady-state waiting-time distribution to have asymptotics of the form x –1 log P(W > x) → –θ ∗as x → ∞for θ ∗ > 0. We require only stationarity of the basic sequence of service times minus interarrival times and a Gärtner–Ellis condition for the cumulant generating function of the associated partial sums, i.e. n –1 log E exp (θSn ) → ψ (θ) as n → ∞, plus regularity conditions on the decay rate function ψ. The asymptotic decay rate θ is the root of the equation ψ (θ) = 0. This result in turn implies a corresponding asymptotic result for the steady-state workload in a queue with general non-decreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates.

scholarly journals On A Single Server Queue with Two-Stage First Essential Service Followed by One of the Two Types of Additional Optional Service and Optional Deterministic Server Vacations

2021 ◽  
Vol 15 ◽  
pp. 1730-1736
Author(s):  
Kailash C. Madan
Keyword(s):  
Steady State ◽  
Single Server ◽  
Single Server Queue ◽  
Optional Service ◽  
General Service Times ◽  
Server Queue ◽  
Service Times ◽  
Two Stages ◽  
General Service ◽  
Essential Service

We study the steady state behavior of a batch arrival single server queue in which the first service consisting of two stages with general service times G1 and G2 is compulsory. After completion of the two stages of the first essential service, a customer has the option of choosing one of the two types of additional service with respective general service times G1 and G2 . Just after completing both stages of first essential service with or without one of the two types of additional optional service, the server has the choice of taking an optional deterministic vacation of fixed (constant) length of time. We obtain steady state probability generating functions for the queue size for various states of the system at a random epoch of time in explicit and closed forms. The steady state results of some interesting special cases have been derived from the main results.

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