The busy period of the E k/G/1 queue by finite waiting room

1975 ◽  
Vol 7 (02) ◽  
pp. 416-430
Author(s):  
A. L. Truslove
Keyword(s):  
Markov Chain ◽  
Busy Period ◽  
Waiting Room ◽  
Queueing Process ◽  
Number Of Customers

For the E k /G/1 queue with finite waiting room the phase technique is used to analyse the Markov chain imbedded in the queueing process at successive instants at which customers complete service, and the distribution of the busy period, together with the number of customers who arrive, and the number of customers served, during that period, is obtained. The limit as the size of the waiting room becomes infinite is found.

The busy period of the Ek/G/1 queue by finite waiting room

1975 ◽  
Vol 7 (2) ◽  
pp. 416-430 ◽  
Author(s):  
A. L. Truslove
Keyword(s):  
Markov Chain ◽  
Busy Period ◽  
Waiting Room ◽  
Queueing Process ◽  
Number Of Customers

For the Ek/G/1 queue with finite waiting room the phase technique is used to analyse the Markov chain imbedded in the queueing process at successive instants at which customers complete service, and the distribution of the busy period, together with the number of customers who arrive, and the number of customers served, during that period, is obtained. The limit as the size of the waiting room becomes infinite is found.

Queue length for the E k/G/1 queue by finite waiting room

1975 ◽  
Vol 7 (01) ◽  
pp. 215-226
Author(s):  
A. L. Truslove
Keyword(s):  
Markov Chain ◽  
Queue Length ◽  
Waiting Room ◽  
Queueing Process

For the E k /G/1 queue with finite waiting room the phase technique is used to analyse the Markov chain imbedded in the queueing process at successive instants at which customers complete service, and the distribution of queue length is obtained. The limit as the size of the waiting room becomes infinite is found.

On the busy periods for the M/G/1 queue with finite and with infinite waiting room

1971 ◽  
Vol 8 (4) ◽  
pp. 821-827 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Joint Distribution ◽  
Busy Period ◽  
Stieltjes Transform ◽  
Waiting Room ◽  
Number Of Customers

SummaryThe Laplace-Stieltjes transform of the distribution of the busy period for the M/G/1 system with infinite waiting room can be obtained by using an argument from branching theory. In the present paper it is shown that by applying this argument it is rather easy to derive the expression for the joint distribution of the busy period and the maximum number of customers present simultaneously during this busy period for the M/G/1 system with infinite waiting room as well as the expression for the distribution of the busy period for the M/G/1 system with finite waiting room.

Queue length for the Ek/G/1 queue by finite waiting room

1975 ◽  
Vol 7 (1) ◽  
pp. 215-226 ◽  
Author(s):  
A. L. Truslove
Keyword(s):  
Markov Chain ◽  
Queue Length ◽  
Waiting Room ◽  
Queueing Process

For the Ek/G/1 queue with finite waiting room the phase technique is used to analyse the Markov chain imbedded in the queueing process at successive instants at which customers complete service, and the distribution of queue length is obtained. The limit as the size of the waiting room becomes infinite is found.

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.

scholarly journals Hitting times of Markov chains, with application to state-dependent queues

1977 ◽  
Vol 17 (1) ◽  
pp. 97-107 ◽  
Author(s):  
R.L. Tweedie
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Busy Period ◽  
Hitting Times ◽  
Arrival Process ◽  
General State ◽  
State Dependent ◽  
General State Space ◽  
The Mean ◽  
Number Of Customers

We present in this note a useful extension of the criteria given in a recent paper [Advances in Appl. Probability 8 (1976), 737–771] for the finiteness of hitting times and mean hitting times of a Markov chain on sets in its (general) state space. We illustrate our results by giving conditions for the finiteness of the mean number of customers in the busy period of a queue in which both the service-times and the arrival process may depend on the waiting time in the queue. Such conditions also suffice for the embedded waiting time chain to have a unique stationary distribution.

On the busy periods for the M/G/1 queue with finite and with infinite waiting room

1971 ◽  
Vol 8 (04) ◽  
pp. 821-827 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Joint Distribution ◽  
Busy Period ◽  
Stieltjes Transform ◽  
Waiting Room ◽  
Number Of Customers

Summary The Laplace-Stieltjes transform of the distribution of the busy period for the M/G/1 system with infinite waiting room can be obtained by using an argument from branching theory. In the present paper it is shown that by applying this argument it is rather easy to derive the expression for the joint distribution of the busy period and the maximum number of customers present simultaneously during this busy period for the M/G/1 system with infinite waiting room as well as the expression for the distribution of the busy period for the M/G/1 system with finite waiting room.

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 M/G/1 vacation model with limited service discipline and hybrid switching-on policy

1997 ◽  
Vol 3 (3) ◽  
pp. 243-253
Author(s):  
Alexander V. Babitsky
Keyword(s):  
Generating Function ◽  
Busy Period ◽  
Optimization Problem ◽  
Queueing System ◽  
Service Discipline ◽  
Multiple Vacations ◽  
Queueing Process ◽  
Vacation Model ◽  
Hybrid Switching ◽  
Limited Service

The author studies an M/G/1 queueing system with multiple vacations. The server is turned off in accordance with the K-limited discipline, and is turned on in accordance with the T-N-hybrid policy. This is to say that the server will begin a vacation from the system if either the queue is empty orKcustomers were served during a busy period. The server idles until it finds at leastNwaiting units upon return from a vacation.Formulas for the distribution generating function and some characteristics of the queueing process are derived. An optimization problem is discussed.

The cyclic queue with one general and one exponential server

1983 ◽  
Vol 15 (04) ◽  
pp. 857-873 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Time Distribution ◽  
Joint Distribution ◽  
Present Model ◽  
Response Times ◽  
Strong Connection ◽  
Waiting Room ◽  
Successive Response ◽  
Proportional Rate ◽  
The Mean ◽  
Number Of Customers

This paper considers the two-stage cyclic queueing model consisting of one general (G) and one exponential (M) server. The strong connection between the present model and the M/G/1 model (with finite waiting room) is exploited to yield the joint distribution of the successive response times of a customer at the G queue and the M queue. This result reveals a surprising phenomenon: in general there is a difference between the joint distribution of the two successive response times at (first) the G queue and (then) the M queue, and the joint distribution of the two successive response times at (first) the M queue and (then) the G queue. Another associated result is an expression for the cycle-time distribution. Special consideration is given to the case that the number of customers in the system tends to ∞, while the mean service times tend to 0 at an inversely proportional rate.

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