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.

Busy period in GIX/G/∞

1996 ◽  
Vol 33 (03) ◽  
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.

Distribution of the busy period in a controllable M/M/2 queue operating under the triadic (0, K, N, M) policy

1990 ◽  
Vol 27 (02) ◽  
pp. 425-432
Author(s):  
Hahn-Kyou Rhee ◽  
B. D. Sivazlian
Keyword(s):  
Steady State ◽  
Busy Period ◽  
Queueing System ◽  
Service Completion ◽  
Special Cases ◽  
Service Stations ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Number Of Customers ◽  
System Operating

We consider an M/M/2 queueing system with removable service stations operating under steady-state conditions. We assume that the number of operating service stations can be adjusted at customers' arrival or service completion epochs depending on the number of customers in the system. The objective of this paper is to obtain the distribution of the busy period using the theory of the gambler's ruin problem. As special cases, the distributions of the busy periods for the ordinary M/M/2 queueing system, the M/M/1 queueing system operating under the N policy and the ordinary M/M/1 queueing system are obtained.

scholarly journals Explicit results for the distribution of the number of customers served during a busy period for $M^X/PH/1$ queue

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Veena Goswami ◽  
M. L. Chaudhry
Keyword(s):  
Functional Equation ◽  
Busy Period ◽  
Stochastic Modelling ◽  
Lagrange Inversion ◽  
Special Cases ◽  
Service Distribution ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Laplace Transform ◽  
Number Of Customers

<p style='text-indent:20px;'>We give analytically explicit solutions for the distribution of the number of customers served during a busy period for the <inline-formula><tex-math id="M1">\begin{document}$ M^X/PH/1 $\end{document}</tex-math></inline-formula> queues when initiated with <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> customers. When customers arrive in batches, we present the functional equation for the Laplace transform of the number of customers served during a busy period. Applying the Lagrange inversion theorem, we provide a refined result to this functional equation. From a phase-type service distribution, we obtain the distribution of the number of customers served during a busy period for various special cases such as exponential, Erlang-k, generalized Erlang, hyperexponential, Coxian, and interrupted Poisson process. The results are exact, rapid and vigorous, owing to the clarity of the expressions. Moreover, we also consider computational results for several service-time distributions using our method. Phase-type distributions can approximate any non-negative valued distribution arbitrarily close, making them a useful practical stochastic modelling tool. These distributions have eloquent properties which make them beneficial in the computation of performance models.</p>

INFINITE-SERVER QUEUES WITH SYSTEM'S ADDITIONAL TASKS AND IMPATIENT CUSTOMERS

2008 ◽  
Vol 22 (4) ◽  
pp. 477-493 ◽  
Author(s):  
Eitan Altman ◽  
Uri Yechiali
Keyword(s):  
Quality Of Service ◽  
Generating Functions ◽  
Busy Period ◽  
Impatient Customers ◽  
Partial Probability ◽  
The Mean ◽  
Infinite Server Queues ◽  
Mean Cycle ◽  
Number Of Customers

A system is operating as an M/M/∞ queue. However, when it becomes empty, it is assigned to perform another task, the duration U of which is random. Customers arriving while the system is unavailable for service (i.e., occupied with a U-task) become impatient: Each individual activates an “impatience timer” having random duration T such that if the system does not become available by the time the timer expires, the customer leaves the system never to return. When the system completes a U-task and there are waiting customers, each one is taken immediately into service. We analyze both multiple and single U-task scenarios and consider both exponentially and generally distributed task and impatience times. We derive the (partial) probability generating functions of the number of customers present when the system is occupied with a U-task as well as when it acts as an M/M/∞ queue and we obtain explicit expressions for the corresponding mean queue sizes. We further calculate the mean length of a busy period, the mean cycle time, and the quality of service measure: proportion of customers being served.

An explicit upper bound for the mean busy period in a GI/G/1 queue

1978 ◽  
Vol 15 (2) ◽  
pp. 452-455 ◽  
Author(s):  
Richard Loulou
Keyword(s):  
Random Walk ◽  
Upper Bound ◽  
Busy Period ◽  
Random Variables ◽  
Cycle Duration ◽  
Traffic Intensity ◽  
The Mean ◽  
Random Walk Theory ◽  
Busy Cycle

In this paper, an upper bound is derived for the mean busy cycle duration in GI/G/1 queues. The bound is of the form A/(1 – ρ), where ρ is the traffic intensity and A involves three moments of the basic random variables of the queue. The proof uses a well-known result of random walk theory.

Distribution of the busy period in a controllable M/M/2 queue operating under the triadic (0, K, N, M) policy

1990 ◽  
Vol 27 (2) ◽  
pp. 425-432 ◽  
Author(s):  
Hahn-Kyou Rhee ◽  
B. D. Sivazlian
Keyword(s):  
Steady State ◽  
Busy Period ◽  
Queueing System ◽  
Service Completion ◽  
Special Cases ◽  
Service Stations ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Number Of Customers ◽  
System Operating

We consider an M/M/2 queueing system with removable service stations operating under steady-state conditions. We assume that the number of operating service stations can be adjusted at customers' arrival or service completion epochs depending on the number of customers in the system. The objective of this paper is to obtain the distribution of the busy period using the theory of the gambler's ruin problem. As special cases, the distributions of the busy periods for the ordinary M/M/2 queueing system, the M/M/1 queueing system operating under the N policy and the ordinary M/M/1 queueing system are obtained.

An explicit upper bound for the mean busy period in a GI/G/1 queue

1978 ◽  
Vol 15 (02) ◽  
pp. 452-455 ◽  
Author(s):  
Richard Loulou
Keyword(s):  
Random Walk ◽  
Upper Bound ◽  
Busy Period ◽  
Random Variables ◽  
Cycle Duration ◽  
Traffic Intensity ◽  
The Mean ◽  
Random Walk Theory ◽  
Busy Cycle

In this paper, an upper bound is derived for the mean busy cycle duration in GI/G/1 queues. The bound is of the form A/(1 – ρ), where ρ is the traffic intensity and A involves three moments of the basic random variables of the queue. The proof uses a well-known result of random walk theory.

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 Analytically Explicit Results for the Distribution of the Number of Customers Served during a Busy Period for Special Cases of the M/G/1 Queue

2019 ◽  
Vol 2019 ◽  
pp. 1-15
Author(s):  
M. L. Chaudhry ◽  
Veena Goswami
Keyword(s):  
Poisson Process ◽  
Busy Period ◽  
Inverse Gaussian ◽  
Chi Square ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Phase Type ◽  
The Laplace Transform ◽  
Markov Modulated ◽  
Number Of Customers

This paper presents analytically explicit results for the distribution of the number of customers served during a busy period for special cases of the M/G/1 queues when initiated with m customers. The functional equation for the Laplace transform of the number of customers served during a busy period is widely known, but several researchers state that, in general, it is not easy to invert it except for some simple cases such as M/M/1 and M/D/1 queues. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We obtain the distribution of the number of customers served during a busy period for various service-time distributions such as exponential, deterministic, Erlang-k, gamma, chi-square, inverse Gaussian, generalized Erlang, matrix exponential, hyperexponential, uniform, Coxian, phase-type, Markov-modulated Poisson process, and interrupted Poisson process. Further, we also provide computational results using our method. The derivations are very fast and robust due to the lucidity of the expressions.

ANALYTICALLY CLOSED-FORM SOLUTIONS FOR THE DISTRIBUTION OF A NUMBER OF CUSTOMERS SERVED DURING A BUSY PERIOD FOR SPECIAL CASES OF THE GEO/G/1 QUEUE

2020 ◽  
pp. 1-30
Author(s):  
M. L. Chaudhry ◽  
Veena Goswami ◽  
Abdalla Mansur
Keyword(s):  
Busy Period ◽  
Negative Binomial ◽  
Closed Form Solutions ◽  
Early Arrival ◽  
Special Cases ◽  
Discrete Phase ◽  
Mean And Variance ◽  
The Mean ◽  
Limiting Case ◽  
Number Of Customers

Abstract This paper presents the distribution of the number of customers served during a busy period for special cases of the Geo/G/1 queue when initiated with m customers. We analyze the system under the assumptions of a late arrival system with delayed access and early arrival system policies. It is not easy to invert the functional equation for the number of customers served during a busy period except for the simple case Geo/Geo/1 queue, as stated by several researchers. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We find the distribution of the number of customers served during a busy period for various service-time distributions such as geometric, deterministic, binomial, negative binomial, uniform, Delaporte, discrete phase-type and interrupted Bernoulli process. We compute the mean and variance of these distributions and also give numerical results. Due to the clarity of the expressions, the computations are very fast and robust. We also show that in the limiting case, the results tend to the analogous continuous-time counterparts.

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