The Stationary Waiting Time Distribution for a Single-Server Queue

1965 ◽  
Vol 13 (4) ◽  
pp. 966-976 ◽  
Author(s):  
Alan G. Konheim
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Server Queue ◽  
Stationary Waiting Time

scholarly journals An M/G/1 queue with multiple types of feedback and gated vacations

1997 ◽  
Vol 34 (03) ◽  
pp. 773-784 ◽  
Author(s):  
Onno J. Boxma ◽  
Uri Yechiali
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queue Length ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Server Queue ◽  
Poisson Arrivals

This paper considers a single-server queue with Poisson arrivals and multiple customer feedbacks. If the first service attempt of a newly arriving customer is not successful, he returns to the end of the queue for another service attempt, with a different service time distribution. He keeps trying in this manner (as an ‘old' customer) until his service is successful. The server operates according to the ‘gated vacation' strategy; when it returns from a vacation to find K (new and old) customers, it renders a single service attempt to each of them and takes another vacation, etc. We study the joint queue length process of new and old customers, as well as the waiting time distribution of customers. Some extensions are also discussed.

Stochastic bounds for the single-server queue

1976 ◽  
Vol 80 (3) ◽  
pp. 521-525 ◽  
Author(s):  
J. Köllerström
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Infinite Divisibility ◽  
Single Server ◽  
Traffic Intensity ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Exponential Tail ◽  
Server Queue ◽  
Fitting In

Various elegant properties have been found for the waiting time distribution G for the queue GI/G/1 in statistical equilibrium, such as infinite divisibility ((1), p. 282) and that of having an exponential tail ((11), (2), p. 411, (1), p. 324). Here we derive another property which holds quite generally, provided the traffic intensity ρ < 1, and which is extremely simple, fitting in with the above results as well as yielding some useful properties in the form of upper and lower stochastic bounds for G which augment the bounds obtained by Kingman (5), (6), (8) and by Ross (10).

scholarly journals An M/G/1 queue with multiple types of feedback and gated vacations

1997 ◽  
Vol 34 (3) ◽  
pp. 773-784 ◽  
Author(s):  
Onno J. Boxma ◽  
Uri Yechiali
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queue Length ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Server Queue ◽  
Poisson Arrivals

This paper considers a single-server queue with Poisson arrivals and multiple customer feedbacks. If the first service attempt of a newly arriving customer is not successful, he returns to the end of the queue for another service attempt, with a different service time distribution. He keeps trying in this manner (as an ‘old' customer) until his service is successful. The server operates according to the ‘gated vacation' strategy; when it returns from a vacation to find K (new and old) customers, it renders a single service attempt to each of them and takes another vacation, etc. We study the joint queue length process of new and old customers, as well as the waiting time distribution of customers. Some extensions are also discussed.

Exponential expansion for the tail of the waiting-time probability in the single-server queue with batch arrivals

1988 ◽  
Vol 20 (4) ◽  
pp. 880-895 ◽  
Author(s):  
J. C. W. Van Ommeren
Keyword(s):  
Waiting Time ◽  
Interarrival Time ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Batch Arrivals ◽  
Exponential Expansion ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Server Queue

This paper deals with the single-server queue with batch arrivals. We show that under suitable conditions the waiting-time distribution of an individual customer has an asymptotically exponential expansion. Computationally useful characterizations of the amplitude factor and the decay parameter are given for the practically important case in which the interarrival time and the service time have phase-type distributions.

Exponential approximation of waiting time and queue size for queues in heavy traffic

1990 ◽  
Vol 22 (1) ◽  
pp. 230-240 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Waiting Time ◽  
Wide Class ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Single Server ◽  
Queue Size ◽  
Exponential Approximation ◽  
Waiting Time Distribution ◽  
Stationary Waiting Time ◽  
Stationary Representation

An exponential approximation for the stationary waiting time distribution and the stationary queue size distribution for single-server queues in heavy traffic is given for a wide class of queues. This class contains for example not only queues for which the generic sequence, i.e. the sequence of service times and interarrival times, is stationary but also such queues for which the generic sequence is asymptotically stationary in some sense. The conditions ensuring the exponential approximation of the characteristics considered in heavy traffic are expressed in terms of the invariance principle for the stationary representation of the generic sequence and its first two moments.

Exponential approximation of waiting time and queue size for queues in heavy traffic

1990 ◽  
Vol 22 (01) ◽  
pp. 230-240 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Waiting Time ◽  
Wide Class ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Single Server ◽  
Queue Size ◽  
Exponential Approximation ◽  
Waiting Time Distribution ◽  
Stationary Waiting Time ◽  
Stationary Representation

An exponential approximation for the stationary waiting time distribution and the stationary queue size distribution for single-server queues in heavy traffic is given for a wide class of queues. This class contains for example not only queues for which the generic sequence, i.e. the sequence of service times and interarrival times, is stationary but also such queues for which the generic sequence is asymptotically stationary in some sense. The conditions ensuring the exponential approximation of the characteristics considered in heavy traffic are expressed in terms of the invariance principle for the stationary representation of the generic sequence and its first two moments.

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.

On a single-server queue with group arrivals

1976 ◽  
Vol 13 (3) ◽  
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.

Exponential expansion for the tail of the waiting-time probability in the single-server queue with batch arrivals

1988 ◽  
Vol 20 (04) ◽  
pp. 880-895 ◽  
Author(s):  
J. C. W. Van Ommeren
Keyword(s):  
Waiting Time ◽  
Interarrival Time ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Batch Arrivals ◽  
Exponential Expansion ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Server Queue

This paper deals with the single-server queue with batch arrivals. We show that under suitable conditions the waiting-time distribution of an individual customer has an asymptotically exponential expansion. Computationally useful characterizations of the amplitude factor and the decay parameter are given for the practically important case in which the interarrival time and the service time have phase-type distributions.

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