The probability of large queue lengths and waiting times in a heterogeneous multiserver queue II: Positive recurrence and logarithmic limits

1995 ◽  
Vol 27 (2) ◽  
pp. 567-583 ◽  
Author(s):  
John S. Sadowsky
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Waiting Times ◽  
General Setting ◽  
Batch Arrival ◽  
Positive Recurrence ◽  
Waiting Time Distributions ◽  
Harris Recurrence ◽  
Stationary Queue Length ◽  
Queue Lengths

We continue our investigation of the batch arrival-heterogeneous multiserver queue begun in Part I. In a general setting we prove the positive Harris recurrence of the system, and with no additional conditions we prove logarithmic tail limits for the stationary queue length and waiting time distributions.

The probability of large queue lengths and waiting times in a heterogeneous multiserver queue II: Positive recurrence and logarithmic limits

1995 ◽  
Vol 27 (02) ◽  
pp. 567-583 ◽  
Author(s):  
John S. Sadowsky
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Waiting Times ◽  
General Setting ◽  
Batch Arrival ◽  
Positive Recurrence ◽  
Waiting Time Distributions ◽  
Harris Recurrence ◽  
Stationary Queue Length ◽  
Queue Lengths

We continue our investigation of the batch arrival-heterogeneous multiserver queue begun in Part I. In a general setting we prove the positive Harris recurrence of the system, and with no additional conditions we prove logarithmic tail limits for the stationary queue length and waiting time distributions.

Some general results for many server queues

1973 ◽  
Vol 5 (01) ◽  
pp. 153-169 ◽  
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.

scholarly journals Decomposition of M/M/1 With Unreliable Service and a Working Vacation

2020 ◽  
Vol 9 (1) ◽  
pp. 63
Author(s):  
Joshua Patterson ◽  
Andrzej Korzeniowski
Keyword(s):  
Closed Form ◽  
Failure Rate ◽  
Waiting Time ◽  
Queue Length ◽  
Service Failure ◽  
Working Vacation ◽  
Stationary Queue Length ◽  
Stationary Waiting Time ◽  
Relationship Of ◽  
The Relationship

We use the stationary distribution for the M/M/1 with Unreliable Service and aWorking Vacation (M/M/1/US/WV) given explicitly in (Patterson & Korzeniowski, 2019) to find a decomposition of the stationary queue length N. By applying the distributional form of Little's Law the Laplace-tieltjes Transform of the stationary customer waiting time W is derived. The closed form of the expected value and variance for both N and W is found and the relationship of the expected stationary waiting time as a function of the service failure rate is determined.

Some general results for many server queues

1973 ◽  
Vol 5 (1) ◽  
pp. 153-169 ◽  
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.

Queue length and waiting time analysis of a batch arrival queue with bilevel control

1998 ◽  
Vol 25 (3) ◽  
pp. 191-205 ◽  
Author(s):  
H.W. Lee ◽  
J.G. Park ◽  
B.K. Kim ◽  
S.H. Yoon ◽  
B.Y Ahn ◽  
...  
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Batch Arrival ◽  
Time Analysis

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.

Properties and Moving Time Average for Lévy Walks with Power-Law Waiting-Time Distributions

2014 ◽  
Vol 580-583 ◽  
pp. 3079-3082
Author(s):  
Kai Ying Deng ◽  
Jing Wei Deng
Keyword(s):  
Power Law ◽  
Waiting Time ◽  
Waiting Times ◽  
Time Cost ◽  
Waiting Time Distributions ◽  
Time Average

Lévy walks are a natural model for the description of sub-ballistic, superdiffusive motion. The waiting times and jump lengths of Lévy walks are coupled in the form . The-coupling introduces a time cost for each jump in the form of the generalized velocity , such that long jumps get penalized by a higher time cost. In this paper, we firstly investigate the properties of Lévy walks with power-law waiting-time distributions; then discuss its moving time average.

Stationary queue-length and waiting-time distributions in single-server feedback queues

1984 ◽  
Vol 16 (2) ◽  
pp. 437-446 ◽  
Author(s):  
Ralph L. Disney ◽  
Dieter König ◽  
Volker schmidt
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Arbitrary Point ◽  
Single Server ◽  
Waiting Room ◽  
Bernoulli Feedback ◽  
Stationary Queue Length ◽  
Queueing Processes ◽  
Number Of Customers ◽  
Feedback Time

For M/GI/1/∞ queues with instantaneous Bernoulli feedback time- and customer-stationary characteristics of the number of customers in the system and of the waiting time are investigated. Customer-stationary characteristics are thereby obtained describing the behaviour of the queueing processes, for example, at arrival epochs, at feedback epochs, and at times at which an arbitrary (arriving or fed-back) customer enters the waiting room. The method used to obtain these characteristics consists of simple relationships between them and the time-stationary distribution of the number of customers in the system at an arbitrary point in time. The latter is obtained from the wellknown Pollaczek–Khinchine formula for M/GI/1/∞ queues without feedback.

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.

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