Some extensions of Takács's limit theorems

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

Advances in Applied Probability ◽

10.2307/1426170 ◽

1971 ◽

Vol 3 (2) ◽

pp. 249-268 ◽

Cited By ~ 16

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.

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

Advances in Applied Probability ◽

10.1017/s0001867800026100 ◽

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.

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

Advances in Applied Probability ◽

10.2307/1427589 ◽

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.

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

Advances in Applied Probability ◽

10.1017/s0001867800037939 ◽

1971 ◽

Vol 3 (02) ◽

pp. 249-268 ◽

Cited By ~ 2

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.

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Some general results for many server queues

Advances in Applied Probability ◽

10.1017/s0001867800039008 ◽

1973 ◽

Vol 5 (01) ◽

pp. 153-169 ◽

Cited By ~ 8

Author(s):

J. H. A. De Smit

Keyword(s):

Integral Equations ◽

Waiting Time ◽

Queue Length ◽

Joint Distribution ◽

Waiting Times ◽

Virtual Waiting Time ◽

Server Queue ◽

Many Server Queues ◽

The Many ◽

Busy Cycle

Pollaczek's theory for the many server queue is generalized and extended. Pollaczek (1961) found the distribution of the actual waiting times in the model G/G/s as a solution of a set of integral equations. We give a somewhat more general set of integral equations from which the joint distribution of the actual waiting time and some other random variables may be found. With this joint distribution we can obtain distributions of a number of characteristic quantities, such as the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. For a wide class of many server queues the formal expressions may lead to explicit results.

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Embedded renewal processes in the GI/G/s queue

Journal of Applied Probability ◽

10.2307/3212333 ◽

1972 ◽

Vol 9 (3) ◽

pp. 650-658 ◽

Cited By ~ 35

Author(s):

Ward Whitt

Keyword(s):

Limit Theorems ◽

Sufficient Conditions ◽

Interarrival Time ◽

Renewal Processes ◽

Positive Probability ◽

Functional Limit ◽

Light Traffic ◽

Functional Limit Theorems ◽

Cumulative Processes ◽

Rule Out

The stable GI/G/s queue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.

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On the comparison of waiting times in tandem queues

Journal of Applied Probability ◽

10.1017/s0021900200098491 ◽

1981 ◽

Vol 18 (03) ◽

pp. 707-714 ◽

Cited By ~ 1

Author(s):

Shun-Chen Niu

Keyword(s):

Waiting Time ◽

Queueing System ◽

Waiting Times ◽

Distribution Functions ◽

Partial Ordering ◽

Tandem Queues ◽

Order Relations ◽

Service Distribution ◽

In Series ◽

Definition Of

Using a definition of partial ordering of distribution functions, it is proven that for a tandem queueing system with many stations in series, where each station can have either one server with an arbitrary service distribution or a number of constant servers in parallel, the expected total waiting time in system of every customer decreases as the interarrival and service distributions becomes smaller with respect to that ordering. Some stronger conclusions are also given under stronger order relations. Using these results, bounds for the expected total waiting time in system are then readily obtained for wide classes of tandem queues.

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Queue-Length, Waiting-Time and Service Batch Size Analysis for the Discrete-Time GI/D-MSP$^{\text {(a,b)}}/1/\infty $ Queueing System

Methodology And Computing In Applied Probability ◽

10.1007/s11009-020-09823-9 ◽

2020 ◽

Author(s):

S. K. Samanta ◽

R. Nandi

Keyword(s):

Discrete Time ◽

Waiting Time ◽

Queue Length ◽

Queueing System ◽

Batch Size ◽

Size Analysis

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Periodic queues in heavy traffic

Advances in Applied Probability ◽

10.1017/s0001867800018693 ◽

1989 ◽

Vol 21 (02) ◽

pp. 485-487 ◽

Cited By ~ 1

Author(s):

G. I. Falin

Keyword(s):

Wiener Process ◽

Waiting Time ◽

Diffusion Approximation ◽

Queueing System ◽

Heavy Traffic ◽

Time Process ◽

Virtual Waiting Time ◽

Waiting Time Process ◽

Periodic Queues ◽

Poisson Arrivals

An analytic approach to the diffusion approximation in queueing due to Burman (1979) is applied to the M(t)/G/1/∞ queueing system with periodic Poisson arrivals. We show that under heavy traffic the virtual waiting time process can be approximated by a certain Wiener process with reflecting barrier at 0.

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