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.

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 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.

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.

scholarly journals On the probability that the kth customer finds an M/M/1 queue empty

1992 ◽  
Vol 24 (1) ◽  
pp. 238-239 ◽  
Author(s):  
Harshinder Singh ◽  
Rameshwar D. Gupta
Keyword(s):  
Busy Period ◽  
Number Of Customers ◽  
The Relationship

A result relating the probability that kth customer finds the system empty to the distribution of the number of customers served in a busy period, for an M/M/1 queue, has been obtained. This relationship is similar to the relationship between the probability that the queue is empty at time t and the distribution of the length of the busy period.

Busy period in GIX/G/∞

1996 ◽  
Vol 33 (3) ◽  
pp. 815-829 ◽  
Author(s):  
Liming Liu ◽  
Ding-Hua Shi
Keyword(s):  
Service Time ◽  
Busy Period ◽  
Random Variables ◽  
Batch Service ◽  
Idle Period ◽  
Special Cases ◽  
Infinite Server Queues ◽  
General Relations ◽  
Number Of Customers ◽  
Busy Cycle

Busy period problems in infinite server queues are studied systematically, starting from the batch service time. General relations are given for the lengths of the busy cycle, busy period and idle period, and for the number of customers served in a busy period. These relations show that the idle period is the most difficult while the busy cycle is the simplest of the four random variables. Renewal arguments are used to derive explicit results for both general and special cases.

scholarly journals Analysis of a Single-Sever Queue with Disasters and Repairs Under Bernoulli Vacation Schedule

2016 ◽  
Vol 4 (6) ◽  
pp. 547-559
Author(s):  
Jingjing Ye ◽  
Liwei Liu ◽  
Tao Jiang
Keyword(s):  
Performance Measures ◽  
Generating Functions ◽  
Sojourn Time ◽  
Time Distribution ◽  
Busy Period ◽  
Balance Equations ◽  
Sojourn Time Distribution ◽  
The Mean ◽  
System Length ◽  
Number Of Customers

AbstractThis paper studies a single-sever queue with disasters and repairs, in which after each service completion the server may take a vacation with probabilityq(0≤q≤1), or begin to serve the next customer, if any, with probabilityp(= 1− q). The disaster only affects the system when the server is in operation, and once it occurs, all customers present are eliminated from the system. We obtain the stationary probability generating functions (PGFs) of the number of customers in the system by solving the balance equations of the system. Some performance measures such as the mean system length, the probability that the server is in different states, the rate at which disasters occur and the rate of initiations of busy period are determined. We also derive the sojourn time distribution and the mean sojourn time. In addition, some numerical examples are presented to show the effect of the parameters on the mean system length.

A new informative embedded Markov renewal process for the PH/G/1 queue

1986 ◽  
Vol 18 (02) ◽  
pp. 533-557 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Statistical Analysis ◽  
Queue Length ◽  
Busy Period ◽  
Waiting Times ◽  
Arrival Process ◽  
Embedded Markov Chain ◽  
Markov Renewal Process ◽  
The Mean ◽  
Busy Cycle

We consider a new embedded Markov chain for the PH/G/1 queue by recording the queue length, the phase of the arrival process and the number of services completed during the current busy period at the successive departure epochs. Algorithmically tractable matrix formulas are obtained which permit the analysis of the fluctuations of the queue length and waiting times during a typical busy cycle. These are useful in the computation of certain profile curves arising in the statistical analysis of queues. In addition, informative expressions for the mean waiting times in the stable GI/G/1 queue and a simple new algorithm to evaluate the waiting-time distributions for the stationary PH/PH/1 queue are obtained.

A note on losses in M/GI/1/n queues

1999 ◽  
Vol 36 (4) ◽  
pp. 1240-1243 ◽  
Author(s):  
Rhonda Righter
Keyword(s):  
Simple Proof ◽  
Service Time ◽  
Busy Period ◽  
Queueing System ◽  
Interarrival Time ◽  
The Mean

Let Ln be the number of losses during a busy period of an M/GI/1/n queueing system. We develop a coupling between Ln and Ln+1 and use the resulting relationship to provide a simple proof that when the mean service time equals the mean interarrival time, ELn = 1 for all n. We also show that Ln is increasing in the convex sense when the mean service time equals the mean interarrival time, and it is increasing in the increasing convex sense when the mean service time is less than the mean interarrival time.

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.

Sign in / Sign up

Export Citation Format

Share Document