Conditioned Limit Theorems for the Difference of Waiting Time and Queue Length

1994 ◽  
Vol 26 (01) ◽  
pp. 242-257
Author(s):  
Władysław Szczotka ◽  
Krzysztof Topolski
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Limit Theorems ◽  
Queueing System ◽  
Queueing Systems ◽  
Traffic Intensity ◽  
Virtual Waiting Time ◽  
The Difference ◽  
Scheme Of Series ◽  
Brownian Meander

Consider the GI/G/1 queueing system with traffic intensity 1 and let wk and lk denote the actual waiting time of the kth unit and the number of units present in the system at the kth arrival including the kth unit, respectively. Furthermore let τ denote the number of units served during the first busy period and μ the intensity of the service. It is shown that as k →∞, where a is some known constant, , , and are independent, is a Brownian meander and is a Wiener process. A similar result is also given for the difference of virtual waiting time and queue length processes. These results are also extended to a wider class of queueing systems than GI/G/1 queues and a scheme of series of queues.

Conditioned Limit Theorems for the Difference of Waiting Time and Queue Length

1994 ◽  
Vol 26 (1) ◽  
pp. 242-257
Author(s):  
Władysław Szczotka ◽  
Krzysztof Topolski
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Limit Theorems ◽  
Queueing System ◽  
Queueing Systems ◽  
Traffic Intensity ◽  
Virtual Waiting Time ◽  
The Difference ◽  
Scheme Of Series ◽  
Brownian Meander

Consider the GI/G/1 queueing system with traffic intensity 1 and let wk and lk denote the actual waiting time of the kth unit and the number of units present in the system at the kth arrival including the kth unit, respectively. Furthermore let τ denote the number of units served during the first busy period and μ the intensity of the service. It is shown that as k →∞, where a is some known constant, , , and are independent, is a Brownian meander and is a Wiener process. A similar result is also given for the difference of virtual waiting time and queue length processes. These results are also extended to a wider class of queueing systems than GI/G/1 queues and a scheme of series of queues.

The virtual waiting time of the GI/G/1 queue in heavy traffic

1971 ◽  
Vol 3 (2) ◽  
pp. 249-268 ◽  
Author(s):  
E. Kyprianou
Keyword(s):  
Markov Chains ◽  
Waiting Time ◽  
Limit Theorems ◽  
Queueing System ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Double Sequence ◽  
Traffic Intensity ◽  
Virtual Waiting Time

Investigations in the theory of heavy traffic were initiated by Kingman ([5], [6] and [7]) in an effort to obtain approximations for stable queues. He considered the Markov chains {Wni} of a sequence {Qi} of stable GI/G/1 queues, where Wni is the waiting time of the nth customer in the ith queueing system, and by making use of Spitzer's identity obtained limit theorems as first n → ∞ and then ρi ↑ 1 as i → ∞. Here &rHi is the traffic intensity of the ith queueing system. After Kingman the theory of heavy traffic was developed by a number of Russians mainly. Prohorov [10] considered the double sequence of waiting times {Wni} and obtained limit theorems in the three cases when n1/2(ρi-1) approaches (i) - ∞, (ii) -δ and (iii) 0 as n → ∞ and i → ∞ simultaneously. The case (i) includes the result of Kingman. Viskov [12] also studied the double sequence {Wni} and obtained limits in the two cases when n1/2(ρi − 1) approaches + δ and + ∞ as n → ∞ and i → ∞ simultaneously.

The virtual waiting time of the GI/G/1 queue in heavy traffic

1971 ◽  
Vol 3 (02) ◽  
pp. 249-268 ◽  
Author(s):  
E. Kyprianou
Keyword(s):  
Markov Chains ◽  
Waiting Time ◽  
Limit Theorems ◽  
Queueing System ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Double Sequence ◽  
Traffic Intensity ◽  
Virtual Waiting Time

Investigations in the theory of heavy traffic were initiated by Kingman ([5], [6] and [7]) in an effort to obtain approximations for stable queues. He considered the Markov chains {W n i } of a sequence {Q i } of stable GI/G/1 queues, where W n i is the waiting time of the nth customer in the ith queueing system, and by making use of Spitzer's identity obtained limit theorems as first n → ∞ and then ρ i ↑ 1 as i → ∞. Here &rH i is the traffic intensity of the ith queueing system. After Kingman the theory of heavy traffic was developed by a number of Russians mainly. Prohorov [10] considered the double sequence of waiting times {W n i } and obtained limit theorems in the three cases when n 1/2(ρ i -1) approaches (i) - ∞, (ii) -δ and (iii) 0 as n → ∞ and i → ∞ simultaneously. The case (i) includes the result of Kingman. Viskov [12] also studied the double sequence {W n i } and obtained limits in the two cases when n 1/2(ρ i − 1) approaches + δ and + ∞ as n → ∞ and i → ∞ simultaneously.

On the waiting-time process in a single-line queue with repeated calls

1986 ◽  
Vol 23 (1) ◽  
pp. 185-192 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Queueing System ◽  
Queueing Systems ◽  
Retrial Queue ◽  
Time Process ◽  
Light Traffic ◽  
Repeated Calls ◽  
Waiting Time Process ◽  
Mean Variance

Waiting time in a queueing system is usually measured by a period from the epoch when a subscriber enters the system until the service starting epoch. For repeated orders queueing systems it is natural to measure the waiting time by the number of repeated attempts, R, which have to be made by a blocked primary call customer before the call enters service. We study this problem for the M/M/1/1 retrial queue and derive expressions for mean, variance and generating function of R. Limit theorems are stated for heavy- and light-traffic cases.

A note on the convexity of performance measures of M/M/c queueing systems

1983 ◽  
Vol 20 (04) ◽  
pp. 920-923 ◽  
Author(s):  
Hau Leung Lee ◽  
Morris A. Cohen
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Queueing System ◽  
Queueing Systems ◽  
Arrival Rate ◽  
Control Problems ◽  
Traffic Intensity ◽  
Expected Number ◽  
Expected Waiting Time

Convexity of performance measures of queueing systems is important in solving control problems of multi-facility systems. This note proves that performance measures such as the expected waiting time, expected number in queue, and the Erlang delay formula are convex with respect to the arrival rate or the traffic intensity of the M/M/c queueing system.

On berry–esseen rate for queue length of the GI/G/K system in heavy traffic

1988 ◽  
Vol 25 (03) ◽  
pp. 596-611
Author(s):  
Xing Jin
Keyword(s):  
Limit Theorem ◽  
Waiting Time ◽  
Queue Length ◽  
Queueing System ◽  
Heavy Traffic ◽  
Traffic Intensity ◽  
Main Method ◽  
Number Of Customers

This paper provides Berry–Esseen rate of limit theorem concerning the number of customers in a GI/G/K queueing system observed at arrival epochs for traffic intensity ρ > 1. The main method employed involves establishing several equalities about waiting time and queue length.

On berry–esseen rate for queue length of the GI/G/K system in heavy traffic

1988 ◽  
Vol 25 (3) ◽  
pp. 596-611 ◽  
Author(s):  
Xing Jin
Keyword(s):  
Limit Theorem ◽  
Waiting Time ◽  
Queue Length ◽  
Queueing System ◽  
Heavy Traffic ◽  
Traffic Intensity ◽  
Main Method ◽  
Number Of Customers

This paper provides Berry–Esseen rate of limit theorem concerning the number of customers in a GI/G/K queueing system observed at arrival epochs for traffic intensity ρ > 1. The main method employed involves establishing several equalities about waiting time and queue length.

On the waiting-time process in a single-line queue with repeated calls

1986 ◽  
Vol 23 (01) ◽  
pp. 185-192 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Queueing System ◽  
Queueing Systems ◽  
Retrial Queue ◽  
Time Process ◽  
Light Traffic ◽  
Repeated Calls ◽  
Waiting Time Process ◽  
Mean Variance

Waiting time in a queueing system is usually measured by a period from the epoch when a subscriber enters the system until the service starting epoch. For repeated orders queueing systems it is natural to measure the waiting time by the number of repeated attempts,R, which have to be made by a blocked primary call customer before the call enters service. We study this problem for theM/M/1/1 retrial queue and derive expressions for mean, variance and generating function ofR.Limit theorems are stated for heavy- and light-traffic cases.

Some extensions of Takács's limit theorems

1974 ◽  
Vol 11 (4) ◽  
pp. 752-761 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Limit Theorems ◽  
Queueing System ◽  
Waiting Times ◽  
Interarrival Time ◽  
Positive Probability ◽  
Limiting Distributions ◽  
Waiting Room ◽  
Poisson Arrivals

In this paper, we establish that if an interarrival time exceeds a service time with a positive probability then the queueing system GI/G/s with a finite waiting room always has proper limiting distributions for its characteristics such as queue length, waiting time and the remaining service times of the customers being served. The result remains valid if we consider a GI/G/s system with bounded waiting times. A technique is also given to establish that for a system with Poisson arrivals the limiting distributions of the queueing characteristics at an epoch of arrival and at an arbitrary epoch are identical.

Some extensions of Takács's limit theorems

1974 ◽  
Vol 11 (04) ◽  
pp. 752-761
Author(s):  
D. N. Shanbhag
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Limit Theorems ◽  
Queueing System ◽  
Waiting Times ◽  
Interarrival Time ◽  
Positive Probability ◽  
Limiting Distributions ◽  
Waiting Room ◽  
Poisson Arrivals

In this paper, we establish that if an interarrival time exceeds a service time with a positive probability then the queueing system GI/G/s with a finite waiting room always has proper limiting distributions for its characteristics such as queue length, waiting time and the remaining service times of the customers being served. The result remains valid if we consider a GI/G/s system with bounded waiting times. A technique is also given to establish that for a system with Poisson arrivals the limiting distributions of the queueing characteristics at an epoch of arrival and at an arbitrary epoch are identical.

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