The rate of convergence in limit theorems for service systems with finite queue capacity

1972 ◽  
Vol 9 (01) ◽  
pp. 87-102 ◽  
Author(s):  
Joseph Tomko
Keyword(s):  
Asymptotic Analysis ◽  
Waiting Time ◽  
Remainder Term ◽  
Time Distribution ◽  
Limit Theorems ◽  
Service Systems ◽  
Traffic Intensity ◽  
Waiting Time Distribution ◽  
Randomly Stopped Sums ◽  
Stopped Sums

The paper deals with the asymptotic analysis of waiting time distribution for service systems with finite queue capacity. First an M/M/m system is considered and the rate of approximation is given. Then the case of the M/G/1 system is studied for traffic intensity ρ > 1. In the last section a condition is given under which an estimate can be derived for the remainder term in central limit theorems for randomly stopped sums.

The rate of convergence in limit theorems for service systems with finite queue capacity

1972 ◽  
Vol 9 (1) ◽  
pp. 87-102 ◽  
Author(s):  
Joseph Tomko
Keyword(s):  
Asymptotic Analysis ◽  
Waiting Time ◽  
Remainder Term ◽  
Time Distribution ◽  
Limit Theorems ◽  
Service Systems ◽  
Traffic Intensity ◽  
Waiting Time Distribution ◽  
Randomly Stopped Sums ◽  
Stopped Sums

The paper deals with the asymptotic analysis of waiting time distribution for service systems with finite queue capacity. First an M/M/m system is considered and the rate of approximation is given. Then the case of the M/G/1 system is studied for traffic intensity ρ > 1. In the last section a condition is given under which an estimate can be derived for the remainder term in central limit theorems for randomly stopped sums.

Stochastic bounds for the single-server queue

1976 ◽  
Vol 80 (3) ◽  
pp. 521-525 ◽  
Author(s):  
J. Köllerström
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Infinite Divisibility ◽  
Single Server ◽  
Traffic Intensity ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Exponential Tail ◽  
Server Queue ◽  
Fitting In

Various elegant properties have been found for the waiting time distribution G for the queue GI/G/1 in statistical equilibrium, such as infinite divisibility ((1), p. 282) and that of having an exponential tail ((11), (2), p. 411, (1), p. 324). Here we derive another property which holds quite generally, provided the traffic intensity ρ < 1, and which is extremely simple, fitting in with the above results as well as yielding some useful properties in the form of upper and lower stochastic bounds for G which augment the bounds obtained by Kingman (5), (6), (8) and by Ross (10).

Heavy traffic theory for queues with several servers. II

1979 ◽  
Vol 16 (2) ◽  
pp. 393-401 ◽  
Author(s):  
Julian Köllerström
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Waiting Time ◽  
Error Bounds ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Asymptotic Properties ◽  
Traffic Intensity ◽  
Waiting Time Distribution ◽  
Traffic Theory

The queues being studied here are of the type GI/G/k in statistical equilibrium (with traffic intensity less than one). The exponential limiting formula for the waiting time distribution function in heavy traffic, conjectured by Kingman (1965) and established by Köllerström (1974), is extended here. The asymptotic properties of the moments are investigated as well as further approximations for the characteristic function and error bounds for the limiting foemulae.

Heavy traffic theory for queues with several servers. II

1979 ◽  
Vol 16 (02) ◽  
pp. 393-401 ◽  
Author(s):  
Julian Köllerström
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Waiting Time ◽  
Error Bounds ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Asymptotic Properties ◽  
Traffic Intensity ◽  
Waiting Time Distribution ◽  
Traffic Theory

The queues being studied here are of the type GI/G/k in statistical equilibrium (with traffic intensity less than one). The exponential limiting formula for the waiting time distribution function in heavy traffic, conjectured by Kingman (1965) and established by Köllerström (1974), is extended here. The asymptotic properties of the moments are investigated as well as further approximations for the characteristic function and error bounds for the limiting foemulae.

Asymptotic correlation in a queue

1969 ◽  
Vol 6 (03) ◽  
pp. 573-583 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Characteristic Function ◽  
Stationary State ◽  
Waiting Time ◽  
Time Distribution ◽  
Queueing Systems ◽  
Traffic Intensity ◽  
Waiting Time Distribution ◽  
General Applicability ◽  
The Difference

Let Xt denote the waiting time of customer t in a stationary GI/G/1 queue, with traffic intensity τ; let ρn denote the correlation between Xt and Xt+n. For a rational GI/G/1 queue, in which the distribution of the difference between arrival and service intervals has a rational characteristic function, it is shown that, for large n, ρn is asymptotically proportional to n– 3/2 e –βn , where β and the factor of proportionality are calculable. The asymptotic law n –3/2 e–βn applies also to the approach of the waiting-time distribution to the stationary state in an initially empty rational GI/G/1 queue, and to the correlations in the queueing systems recently analysed by Cohen [1]. Its more general applicability is discussed.

Stochastic bounds for the queue GI/G/1 in heavy traffic

1978 ◽  
Vol 84 (2) ◽  
pp. 361-375 ◽  
Author(s):  
Julian Köllerström
Keyword(s):  
Distribution Function ◽  
Waiting Time ◽  
Error Bounds ◽  
Queueing Theory ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Characteristic Functions ◽  
Traffic Intensity ◽  
Waiting Time Distribution ◽  
Brownian Approximation

It is often of greater practical value to have results about queueing theory which involve probabilities rather than characteristic functions. To quote Kendall (4), section 5, ‘These results of Prabhu, further exploited by himself and Takács, are rapidly raising the Laplacian curtain which has hitherto obscured much of the details of the queue-theoretic scene’. In this paper we derive the exponential limit formula for the equilibrium waiting time distribution function G, for the queue GI/G/1 in heavy traffic, using stochastic bounds which are asymptotically sharp as the traffic intensity (defined below) increases to unity (which has not been done before to the author's knowledge). This formula was derived by Kingman (5), (6) and (9), using characteristic functions, who, in section 9 of the latter paper, stressed the need for improving the precision of the approximation ‘by giving inequalities, bounds for errors, and generally by setting the theory on a more elegant and rigorous basis’. Kingman (6) and (9) also sketched a proof of the same result using a Brownian approximation, which was done in detail by Viskov (18); but here again the same difficulties are present in practical interpretation, error bounds etc.

Asymptotic correlation in a queue

1969 ◽  
Vol 6 (3) ◽  
pp. 573-583 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Characteristic Function ◽  
Stationary State ◽  
Waiting Time ◽  
Time Distribution ◽  
Queueing Systems ◽  
Traffic Intensity ◽  
Waiting Time Distribution ◽  
General Applicability ◽  
The Difference

Let Xt denote the waiting time of customer t in a stationary GI/G/1 queue, with traffic intensity τ; let ρn denote the correlation between Xt and Xt+n. For a rational GI/G/1 queue, in which the distribution of the difference between arrival and service intervals has a rational characteristic function, it is shown that, for large n, ρn is asymptotically proportional to n–3/2e–βn, where β and the factor of proportionality are calculable. The asymptotic law n–3/2e–βn applies also to the approach of the waiting-time distribution to the stationary state in an initially empty rational GI/G/1 queue, and to the correlations in the queueing systems recently analysed by Cohen [1]. Its more general applicability is discussed.

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