Virtual Waiting Time in the Single-server Queuing System with Unreliable Server

2004 ◽  
Vol 65 (12) ◽  
pp. 1968-1976 ◽  
Author(s):  
I. S. Mikadze ◽  
V. V. Khocholava ◽  
R. A. Khurodze
Keyword(s):  
Waiting Time ◽  
Queuing System ◽  
Single Server ◽  
Unreliable Server ◽  
Virtual Waiting Time

WAITING TIME ANALYSIS OF MULTI-CLASS QUEUES WITH IMPATIENT CUSTOMERS

2013 ◽  
Vol 27 (3) ◽  
pp. 333-352 ◽  
Author(s):  
Vahid Sarhangian ◽  
Bariş Balciog̃lu
Keyword(s):  
Waiting Time ◽  
Delay Systems ◽  
Single Server ◽  
Impatient Customers ◽  
Preemptive Priority ◽  
System A ◽  
Virtual Waiting Time ◽  
The Laplace Transform ◽  
Time Requirements ◽  
Priority Policy

In this paper, we study three delay systems where different classes of impatient customers arrive according to independent Poisson processes. In the first system, a single server receives two classes of customers with general service time requirements, and follows a non-preemptive priority policy in serving them. Both classes of customers abandon the system when their exponentially distributed patience limits expire. The second system comprises parallel and identical servers providing the same type of service for both classes of impatient customers under the non-preemptive priority policy. We assume exponential service times and consider two cases depending on the time-to-abandon distribution being exponentially distributed or deterministic. In either case, we permit different reneging rates or patience limits for each class. Finally, we consider the first-come-first-served policy in single- and multi-server settings. In all models, we obtain the Laplace transform of the virtual waiting time for each class by exploiting the level-crossing method. This enables us to compute the steady-state system performance measures.

A single server tandem queue

1971 ◽  
Vol 8 (1) ◽  
pp. 95-109 ◽  
Author(s):  
Sreekantan S. Nair
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Busy Period ◽  
Single Server ◽  
Queueing Model ◽  
Tandem Queue ◽  
Limiting Behavior ◽  
Virtual Waiting Time ◽  
Priority Model

Avi-Itzhak, Maxwell and Miller (1965) studied a queueing model with a single server serving two service units with alternating priority. Their model explored the possibility of having the alternating priority model treated in this paper with a single server serving alternately between two service units in tandem.Here we study the distribution of busy period, virtual waiting time and queue length and their limiting behavior.

Single-server queues with impatient customers

1984 ◽  
Vol 16 (4) ◽  
pp. 887-905 ◽  
Author(s):  
F. Baccelli ◽  
P. Boyer ◽  
G. Hebuterne
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Distribution Functions ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Input Process ◽  
Poisson Input ◽  
Virtual Waiting Time ◽  
Random Threshold ◽  
The Stability

We consider a single-server queueing system in which a customer gives up whenever his waiting time is larger than a random threshold, his patience time. In the case of a GI/GI/1 queue with i.i.d. patience times, we establish the extensions of the classical GI/GI/1 formulae concerning the stability condition and the relation between actual and virtual waiting-time distribution functions. We also prove that these last two distribution functions coincide in the case of a Poisson input process and determine their common law.

On two stationary distributions for the stable GI/G/1 queue

1974 ◽  
Vol 11 (04) ◽  
pp. 849-852 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Distribution Functions ◽  
Single Server ◽  
Stationary Distributions ◽  
Virtual Waiting Time ◽  
Server Systems ◽  
Stationary Queue Length

This paper provides simple proofs of two standard results for the stable GI/G/1 queue on the structure of the distribution functions of the stationary virtual waiting time and the stationary queue-length Our argument is applicable to more general single server systems than the queue GI/G/1.

The N/G/1 queue and its detailed analysis

1980 ◽  
Vol 12 (01) ◽  
pp. 222-261 ◽  
Author(s):  
V. Ramaswami
Keyword(s):  
Poisson Process ◽  
Waiting Time ◽  
Queue Length ◽  
Renewal Theory ◽  
Single Server ◽  
Virtual Waiting Time ◽  
Special Cases ◽  
Complex Models ◽  
Markov Modulated ◽  
Markov Renewal Theory

We discuss a single-server queue whose input is the versatile Markovian point process recently introduced by Neuts [22] herein to be called the N-process. Special cases of the N-process discussed earlier in the literature include a number of complex models such as the Markov-modulated Poisson process, the superposition of a Poisson process and a phase-type renewal process, etc. This queueing model has great appeal in its applicability to real world situations especially such as those involving inhibition or stimulation of arrivals by certain renewals. The paper presents formulas in forms which are computationally tractable and provides a unified treatment of many models which were discussed earlier by several authors and which turn out to be special cases. Among the topics discussed are busy-period characteristics, queue-length distributions, moments of the queue length and virtual waiting time. We draw particular attention to our generalization of the Pollaczek–Khinchin formula for the Laplace–Stieltjes transform of the virtual waiting time of the M/G/1 queue to the present model and the resulting Volterra system of integral equations. The analysis presented here serves as an example of the power of Markov renewal theory.

On two stationary distributions for the stable GI/G/1 queue

1974 ◽  
Vol 11 (4) ◽  
pp. 849-852 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Distribution Functions ◽  
Single Server ◽  
Stationary Distributions ◽  
Virtual Waiting Time ◽  
Server Systems ◽  
Stationary Queue Length

This paper provides simple proofs of two standard results for the stable GI/G/1 queue on the structure of the distribution functions of the stationary virtual waiting time and the stationary queue-length Our argument is applicable to more general single server systems than the queue GI/G/1.

On a single-server queue with group arrivals

1976 ◽  
Vol 13 (03) ◽  
pp. 619-622 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Limiting Distribution ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Regenerative Processes ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Individual Service

The queueing system GI/G/1 with group arrivals and individual service of the customers is considered. For the stable situation the limiting distribution of the waiting time distribution of the kth served customer for k → ∞ is derived by using the theory of regenerative processes. It is assumed that the group sizes are i.i.d. variables of which the distribution is aperiodic. The relation between this limiting distribution and the stationary distribution of the virtual waiting time is derived.

A single server tandem queue

1971 ◽  
Vol 8 (01) ◽  
pp. 95-109
Author(s):  
Sreekantan S. Nair
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Busy Period ◽  
Single Server ◽  
Queueing Model ◽  
Tandem Queue ◽  
Limiting Behavior ◽  
Virtual Waiting Time ◽  
Priority Model

Avi-Itzhak, Maxwell and Miller (1965) studied a queueing model with a single server serving two service units with alternating priority. Their model explored the possibility of having the alternating priority model treated in this paper with a single server serving alternately between two service units in tandem. Here we study the distribution of busy period, virtual waiting time and queue length and their limiting behavior.

The N/G/1 queue and its detailed analysis

1980 ◽  
Vol 12 (1) ◽  
pp. 222-261 ◽  
Author(s):  
V. Ramaswami
Keyword(s):  
Poisson Process ◽  
Waiting Time ◽  
Queue Length ◽  
Renewal Theory ◽  
Single Server ◽  
Virtual Waiting Time ◽  
Special Cases ◽  
Complex Models ◽  
Markov Modulated ◽  
Markov Renewal Theory

We discuss a single-server queue whose input is the versatile Markovian point process recently introduced by Neuts [22] herein to be called the N-process. Special cases of the N-process discussed earlier in the literature include a number of complex models such as the Markov-modulated Poisson process, the superposition of a Poisson process and a phase-type renewal process, etc. This queueing model has great appeal in its applicability to real world situations especially such as those involving inhibition or stimulation of arrivals by certain renewals. The paper presents formulas in forms which are computationally tractable and provides a unified treatment of many models which were discussed earlier by several authors and which turn out to be special cases. Among the topics discussed are busy-period characteristics, queue-length distributions, moments of the queue length and virtual waiting time. We draw particular attention to our generalization of the Pollaczek–Khinchin formula for the Laplace–Stieltjes transform of the virtual waiting time of the M/G/1 queue to the present model and the resulting Volterra system of integral equations. The analysis presented here serves as an example of the power of Markov renewal theory.

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