The Distribution of the Maximum Queue Length, The Number of Customers Served and The Duration of The Busy Period for The Queueing SystemM/M/1Involving Batches

1971 ◽  
Vol 9 (2) ◽  
pp. 161-166 ◽  
Author(s):  
S.G. Mohanty
Keyword(s):  
Queue Length ◽  
Busy Period ◽  
Distribution Of The Maximum ◽  
Maximum Queue Length ◽  
Number Of Customers

The Trivariate Distribution of the Maximum Queue Length, the Number of Customers Served and the Duration of the Busy Period for the M/G/1 Queueing System

1969 ◽  
Vol 6 (1) ◽  
pp. 154-161 ◽  
Author(s):  
E.G. Enns
Keyword(s):  
Queue Length ◽  
Busy Period ◽  
Queueing System ◽  
Single Server ◽  
List Type ◽  
Type Number ◽  
Distribution Of The Maximum ◽  
Maximum Queue Length ◽  
Number Of Customers

In the study of the busy period for a single server queueing system, three variables that have been investigated individually or at most in pairs are:1.The duration of the busy period.2.The number of customers served during the busy period.3.The maximum number of customers in the queue during the busy period.

The Trivariate Distribution of the Maximum Queue Length, the Number of Customers Served and the Duration of the Busy Period for the M/G/1 Queueing System

1969 ◽  
Vol 6 (01) ◽  
pp. 154-161 ◽  
Author(s):  
E.G. Enns
Keyword(s):  
Queue Length ◽  
Busy Period ◽  
Queueing System ◽  
Single Server ◽  
List Type ◽  
Type Number ◽  
Distribution Of The Maximum ◽  
Maximum Queue Length ◽  
Number Of Customers

In the study of the busy period for a single server queueing system, three variables that have been investigated individually or at most in pairs are: 1. The duration of the busy period. 2. The number of customers served during the busy period. 3. The maximum number of customers in the queue during the busy period.

The distribution of the maximum number of customers present simultaneously during a busy period for the queueing systems M/G/1 and G/M/1

1967 ◽  
Vol 4 (01) ◽  
pp. 162-179 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Busy Period ◽  
Transition Probabilities ◽  
Queueing Systems ◽  
Entrance Time ◽  
Distribution Of The Maximum ◽  
Number Of Customers

The distribution of the maximum number of customers simultaneously present during a busy period is studied for the queueing systems M/G/1 and G/M/1. These distributions are obtained by using taboo probabilities. Some new relations for transition probabilities and entrance time distributions are derived.

Algorithmic Analysis of the Maximum Queue Length in a Busy Period for theM/M/cRetrial Queue

2007 ◽  
Vol 19 (1) ◽  
pp. 121-126 ◽  
Author(s):  
Jesus R. Artalejo ◽  
Antonis Economou ◽  
M. J. Lopez-Herrero
Keyword(s):  
Queue Length ◽  
Busy Period ◽  
Maximum Queue Length ◽  
Algorithmic Analysis

scholarly journals The effect of service time variability on maximum queue lengths in MX/G/1 queues

2005 ◽  
Vol 42 (3) ◽  
pp. 883-891 ◽  
Author(s):  
Ger Koole ◽  
Misja Nuyens ◽  
Rhonda Righter
Keyword(s):  
Service Time ◽  
Queue Length ◽  
Busy Period ◽  
Random Variable ◽  
Buffer Size ◽  
Time Variability ◽  
Finite Buffer ◽  
Surprising Result ◽  
Maximum Queue Length ◽  
The Impact

We study the impact of service time distributions on the distribution of the maximum queue length during a busy period for the MX/G/1 queue. The maximum queue length is an important random variable to understand when designing the buffer size for finite-buffer (M/G/1/n) systems. We show the somewhat surprising result that, for three variations of the preemptive last-come–first-served discipline, the maximum queue length during a busy period is smaller when service times are more variable (in the convex sense).

The distribution of the maximum number of customers present simultaneously during a busy period for the queueing systems M/G/1 and G/M/1

1967 ◽  
Vol 4 (1) ◽  
pp. 162-179 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Busy Period ◽  
Transition Probabilities ◽  
Queueing Systems ◽  
Entrance Time ◽  
Distribution Of The Maximum ◽  
Number Of Customers

The distribution of the maximum number of customers simultaneously present during a busy period is studied for the queueing systems M/G/1 and G/M/1. These distributions are obtained by using taboo probabilities. Some new relations for transition probabilities and entrance time distributions are derived.

On the relationship between the distribution of maximal queue length in the M/G/1 queue and the mean busy period in the M/G/1/n queue

1976 ◽  
Vol 13 (1) ◽  
pp. 195-199 ◽  
Author(s):  
Robert B. Cooper ◽  
Borge Tilt
Keyword(s):  
Service Time ◽  
Queue Length ◽  
Busy Period ◽  
Waiting Room ◽  
The Mean ◽  
Number Of Customers ◽  
The Relationship

Takács has shown that, in the M/G/1 queue, the probability P(k | i) that the maximum number of customers present simultaneously during a busy period that begins with i customers present is P(k | i) = Qk–i/Qk, where the Q's are easily calculated by recurrence in terms of an arbitrary Q0 ≠ 0. We augment Takács's theorem by showing that P(k | i) = bk–i/bk, where bn is the mean busy period in the M/G/1 queue with finite waiting room of size n; that is, if we take Q0 equal to the mean service time, then Qn =bn.

On the relationship between the distribution of maximal queue length in the M/G/1 queue and the mean busy period in the M/G/1/n queue

1976 ◽  
Vol 13 (01) ◽  
pp. 195-199 ◽  
Author(s):  
Robert B. Cooper ◽  
Borge Tilt
Keyword(s):  
Service Time ◽  
Queue Length ◽  
Busy Period ◽  
Waiting Room ◽  
The Mean ◽  
Number Of Customers ◽  
The Relationship

Takács has shown that, in the M/G/1 queue, the probability P(k | i) that the maximum number of customers present simultaneously during a busy period that begins with i customers present is P(k | i) = Qk –i /Qk, where the Q's are easily calculated by recurrence in terms of an arbitrary Q 0 ≠ 0. We augment Takács's theorem by showing that P(k | i) = bk –i /bk, where bn is the mean busy period in the M/G/1 queue with finite waiting room of size n; that is, if we take Q 0 equal to the mean service time, then Qn =bn.

scholarly journals The effect of service time variability on maximum queue lengths in MX/G/1 queues

2005 ◽  
Vol 42 (03) ◽  
pp. 883-891
Author(s):  
Ger Koole ◽  
Misja Nuyens ◽  
Rhonda Righter
Keyword(s):  
Service Time ◽  
Queue Length ◽  
Busy Period ◽  
Random Variable ◽  
Buffer Size ◽  
Time Variability ◽  
Finite Buffer ◽  
Surprising Result ◽  
Maximum Queue Length ◽  
The Impact

We study the impact of service time distributions on the distribution of the maximum queue length during a busy period for the M X /G/1 queue. The maximum queue length is an important random variable to understand when designing the buffer size for finite-buffer (M/G/1/n) systems. We show the somewhat surprising result that, for three variations of the preemptive last-come–first-served discipline, the maximum queue length during a busy period is smaller when service times are more variable (in the convex sense).

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