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.

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.

Heavy traffic theory for queues with several servers. I

1974 ◽  
Vol 11 (03) ◽  
pp. 544-552 ◽  
Author(s):  
Julian Köllerström
Keyword(s):  
Asymptotic Formula ◽  
Waiting Time ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Waiting Time Distribution ◽  
Arrival Times ◽  
Renewal Sequence ◽  
Queue Discipline ◽  
Traffic Theory ◽  
Mutually Independent

Queues with several servers are examined here, in which arrivals are assumed to form a renewal sequence and successive service times to be mutually independent and independent of the arrival times. The first-come-first-served queue discipline only is considered. An asymptotic formula for the equilibrium waiting time distribution is obtained under conditions of heavy traffic.

Heavy traffic theory for queues with several servers. I

1974 ◽  
Vol 11 (3) ◽  
pp. 544-552 ◽  
Author(s):  
Julian Köllerström
Keyword(s):  
Asymptotic Formula ◽  
Waiting Time ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Waiting Time Distribution ◽  
Arrival Times ◽  
Renewal Sequence ◽  
Queue Discipline ◽  
Traffic Theory ◽  
Mutually Independent

Queues with several servers are examined here, in which arrivals are assumed to form a renewal sequence and successive service times to be mutually independent and independent of the arrival times. The first-come-first-served queue discipline only is considered. An asymptotic formula for the equilibrium waiting time distribution is obtained under conditions of heavy traffic.

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.

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.

CONTINUOUS-TIME FINANCE AND THE WAITING TIME DISTRIBUTION: MULTIPLE CHARACTERISTIC TIMES

2012 ◽  
Vol 26 (23) ◽  
pp. 1250151 ◽  
Author(s):  
KWOK SAU FA
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Waiting Time ◽  
Continuous Time ◽  
Probability Distribution Function ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Exponential Functions ◽  
Multiple Characteristic ◽  
Sum Of Products

In this paper, we model the tick-by-tick dynamics of markets by using the continuous-time random walk (CTRW) model. We employ a sum of products of power law and stretched exponential functions for the waiting time probability distribution function; this function can fit well the waiting time distribution for BUND futures traded at LIFFE in 1997.

scholarly journals The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1

2002 ◽  
Vol 39 (03) ◽  
pp. 619-629 ◽  
Author(s):  
Gang Uk Hwang ◽  
Bong Dae Choi ◽  
Jae-Kyoon Kim
Keyword(s):  
Discrete Time ◽  
Waiting Time ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Autoregressive Process ◽  
Waiting Time Distribution ◽  
Tail Probabilities ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
Asymptotic Decay Rate

We consider a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtain the actual waiting time distribution and the virtual waiting time distribution. As shown in the analysis, our approach provides a natural numerical algorithm to compute the waiting time distributions, based on the theory of the GI/G/1 queue, and consequently we can easily investigate the effect of the parameters of the DAR(1) on the waiting time distributions. We also derive a simple approximation of the asymptotic decay rate of the tail probabilities for the virtual waiting time in the heavy traffic case.

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