Limit theorems for queueing systems with infinite number of servers and group arrival of requests

2016 ◽  
Vol 71 (6) ◽  
pp. 257-260
Author(s):  
E. A. Chernavskaya
Keyword(s):  
Infinite Number ◽  
Limit Theorems ◽  
Queueing Systems ◽  
Group Arrival

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.

scholarly journals Randomized longest-queue-first scheduling for large-scale buffered systems

2015 ◽  
Vol 47 (04) ◽  
pp. 1015-1038 ◽  
Author(s):  
A. B. Dieker ◽  
T. Suk
Keyword(s):  
Limit Theorems ◽  
Large Scale ◽  
Scheduling Algorithm ◽  
Queueing Systems ◽  
Mean Field ◽  
Diffusion Approximations ◽  
Mean Field Limit ◽  
Multi Scale ◽  
Noteworthy Feature ◽  
Longest Queue First

We develop diffusion approximations for parallel-queueing systems with the randomized longest-queue-first scheduling (LQF) algorithm by establishing new mean-field limit theorems as the number of buffers n → ∞. We achieve this by allowing the number of sampled buffers d = d(n) to depend on the number of buffers n, which yields an asymptotic 'decoupling' of the queue length processes. We show through simulation experiments that the resulting approximation is accurate even for moderate values of n and d(n). To the best of the authors' knowledge, this is the first derivation of diffusion approximations for a queueing system in the large-buffer mean-field regime. Another noteworthy feature of our scaling idea is that the randomized LQF algorithm emulates the LQF algorithm, yet is computationally more attractive. The analysis of the system performance as a function of d(n) is facilitated by the multi-scale nature in our limit theorems: the various processes we study have different space scalings. This allows us to show the trade-off between performance and complexity of the randomized LQF scheduling algorithm.

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.

Heavy traffic limit theorems for a sequence of shortest queueing systems

1995 ◽  
Vol 21 (1-2) ◽  
pp. 217-238 ◽  
Author(s):  
Hanqin Zhang ◽  
Guang -Hui Hsu ◽  
Rongxin Wang
Keyword(s):  
Limit Theorems ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Heavy Traffic Limit

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.

scholarly journals Functional Central Limit Theorems for Occupancies and Missing Mass Process in Infinite Urn Models

2020 ◽  
Author(s):  
Mikhail Chebunin ◽  
Sergei Zuyev
Keyword(s):  
Central Limit ◽  
Discrete Time ◽  
Infinite Number ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Urn Models ◽  
Missing Mass ◽  
Urn Scheme ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

AbstractWe study the infinite urn scheme when the balls are sequentially distributed over an infinite number of urns labeled 1,2,... so that the urn j at every draw gets a ball with probability $$p_j$$ p j , where $$\sum _j p_j=1$$ ∑ j p j = 1 . We prove functional central limit theorems for discrete time and the Poissonized version for the urn occupancies process, for the odd occupancy and for the missing mass processes extending the known non-functional central limit theorems.

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