Another martingale bound on the waiting-time distribution in GI/G/1 queues

1979 ◽  
Vol 16 (2) ◽  
pp. 454-457 ◽  
Author(s):  
Harry H. Tan
Keyword(s):  
Waiting Time ◽  
Upper Bound ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Martingale Approach ◽  
Stationary Waiting Time

A new upper bound on the stationary waiting-time distribution of a GI/G/1 queue is derived following Kingman's martingale approach. This bound is generally stronger than Kingman's upper bound and is sometimes stronger than an upper bound derived by Ross.

Another martingale bound on the waiting-time distribution in GI/G/1 queues

1979 ◽  
Vol 16 (02) ◽  
pp. 454-457 ◽  
Author(s):  
Harry H. Tan
Keyword(s):  
Waiting Time ◽  
Upper Bound ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Martingale Approach ◽  
Stationary Waiting Time

A new upper bound on the stationary waiting-time distribution of a GI/G/1 queue is derived following Kingman's martingale approach. This bound is generally stronger than Kingman's upper bound and is sometimes stronger than an upper bound derived by Ross.

COMPUTATIONAL ANALYSIS OF STATIONARY WAITING-TIME DISTRIBUTIONS OF GIX/R/1 AND GIX/D/1 QUEUES

2005 ◽  
Vol 19 (1) ◽  
pp. 121-140 ◽  
Author(s):  
Mohan L. Chaudhry ◽  
Dae W. Choi ◽  
Kyung C. Chae
Keyword(s):  
Closed Form ◽  
Waiting Time ◽  
Time Distribution ◽  
Computational Analysis ◽  
Rational Functions ◽  
Waiting Time Distribution ◽  
Generalized Distributions ◽  
Analytic Expression ◽  
Stationary Waiting Time ◽  
Stieltjes Transforms

In this article, we obtain, in a unified way, a closed-form analytic expression, in terms of roots of the so-called characteristic equation of the stationary waiting-time distribution for the GIX/R/1 queue, where R denotes the class of distributions whose Laplace–Stieltjes transforms are rational functions (ratios of a polynomial of degree at most n to a polynomial of degree n). The analysis is not restricted to generalized distributions with phases such as Coxian-n (Cn) but also covers nonphase-type distributions such as deterministic (D). In the latter case, we get approximate results. Numerical results are presented only for (1) the first two moments of waiting time and (2) the probability that waiting time is zero. It is expected that the results obtained from the present study should prove to be useful not only for practitioners but also for queuing theorists who would like to test the accuracies of inequalities, bounds, or approximations.

Moments Of the stationary waiting time in the Gi/Ph/1 queue

1988 ◽  
Vol 25 (3) ◽  
pp. 636-641 ◽  
Author(s):  
V. Ramaswami ◽  
D. M. Lucantoni
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Higher Moments ◽  
Waiting Time Distribution ◽  
Stationary Waiting Time ◽  
Recursive Relations

Recursive relations for computing the higher moments of the stationary waiting time distribution in a stable GI/PH/1 queue are derived. These provide an accurate and stable technique to compute these moments.

Moments Of the stationary waiting time in the Gi/Ph/1 queue

1988 ◽  
Vol 25 (03) ◽  
pp. 636-641
Author(s):  
V. Ramaswami ◽  
D. M. Lucantoni
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Higher Moments ◽  
Waiting Time Distribution ◽  
Stationary Waiting Time ◽  
Recursive Relations

Recursive relations for computing the higher moments of the stationary waiting time distribution in a stable GI/PH/1 queue are derived. These provide an accurate and stable technique to compute these moments.

On exponential bounds for the waiting-time distribution function in GI/G/1

1976 ◽  
Vol 13 (2) ◽  
pp. 411-417 ◽  
Author(s):  
R. Bergmann ◽  
D. Stoyan
Keyword(s):  
Distribution Function ◽  
Waiting Time ◽  
Time Distribution ◽  
Arrival Time ◽  
Distribution Functions ◽  
Integral Inequalities ◽  
Arrival Time Distribution ◽  
Waiting Time Distribution ◽  
Exponential Bounds ◽  
Stationary Waiting Time

Exponential bounds for the stationary waiting-time distribution of the type ae–θt are considered. These bounds are obtained by the use of Kingman's method of ‘integral inequalities’. Approximations of Θ and a are given which are useful especially if the service and/or inter-arrival time distribution functions are NBUE or NWUE.

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