A note on the MX/GY/1, K bulk queueing system

1973 ◽  
Vol 10 (04) ◽  
pp. 901-906
Author(s):  
Tapan P. Bagchi ◽  
J. G. C. Templeton
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Present Note ◽  
Stationary Queue Length ◽  
Number Of Customers

Cohen (1969) has studied the transient and stationary queue length distributions for the M/G/1, K queue, with a fixed maximum number of customers, K, in the system at any time. The present note applies Cohen's method to generalize his results to the MX/GY/1, K queue.

A note on the MX/GY/1, K bulk queueing system

1973 ◽  
Vol 10 (4) ◽  
pp. 901-906 ◽  
Author(s):  
Tapan P. Bagchi ◽  
J. G. C. Templeton
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Present Note ◽  
Stationary Queue Length ◽  
Number Of Customers

Cohen (1969) has studied the transient and stationary queue length distributions for the M/G/1, K queue, with a fixed maximum number of customers, K, in the system at any time. The present note applies Cohen's method to generalize his results to the MX/GY/1, K queue.

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.

On berry–esseen rate for queue length of the GI/G/K system in heavy traffic

1988 ◽  
Vol 25 (03) ◽  
pp. 596-611
Author(s):  
Xing Jin
Keyword(s):  
Limit Theorem ◽  
Waiting Time ◽  
Queue Length ◽  
Queueing System ◽  
Heavy Traffic ◽  
Traffic Intensity ◽  
Main Method ◽  
Number Of Customers

This paper provides Berry–Esseen rate of limit theorem concerning the number of customers in a GI/G/K queueing system observed at arrival epochs for traffic intensity ρ > 1. The main method employed involves establishing several equalities about waiting time and queue length.

ANALYSIS OF A BULK QUEUE WITH FAST AND SLOW SERVICE RATES AND MULTIPLE VACATIONS

2005 ◽  
Vol 22 (02) ◽  
pp. 239-260 ◽  
Author(s):  
R. ARUMUGANATHAN ◽  
K. S. RAMASWAMI
Keyword(s):  
Performance Measures ◽  
Queue Length ◽  
Queueing System ◽  
Cost Model ◽  
Probability Generating Function ◽  
Service Rate ◽  
Multiple Vacations ◽  
Bulk Queue ◽  
Service Rates ◽  
Number Of Customers

We analyze a Mx/G(a,b)/1 queueing system with fast and slow service rates and multiple vacations. The server does the service with a faster rate or a slower rate based on the queue length. At a service completion epoch (or) at a vacation completion epoch if the number of customers waiting in the queue is greater than or equal to N (N > b), then the service is rendered at a faster rate, otherwise with a slower service rate. After finishing a service, if the queue length is less than 'a' the server leaves for a vacation of random length. When he returns from the vacation, if the queue length is still less than 'a' he leaves for another vacation and so on until he finally finds atleast 'a' customers waiting for service. After a service (or) a vacation, if the server finds atleast 'a' customers waiting for service say ξ, then he serves a batch of min (ξ, b) customers, where b ≥ a. We derive the probability generating function of the queue size at an arbitrary time. Various performance measures are obtained. A cost model is discussed with a numerical solution.

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 berry–esseen rate for queue length of the GI/G/K system in heavy traffic

1988 ◽  
Vol 25 (3) ◽  
pp. 596-611 ◽  
Author(s):  
Xing Jin
Keyword(s):  
Limit Theorem ◽  
Waiting Time ◽  
Queue Length ◽  
Queueing System ◽  
Heavy Traffic ◽  
Traffic Intensity ◽  
Main Method ◽  
Number Of Customers

This paper provides Berry–Esseen rate of limit theorem concerning the number of customers in a GI/G/K queueing system observed at arrival epochs for traffic intensity ρ > 1. The main method employed involves establishing several equalities about waiting time and queue length.

scholarly journals An alternative approach in finding the stationary queue length distribution of a queueing system with negative customers

2021 ◽  
Vol 36 ◽  
pp. 04001
Author(s):  
Siew Khew Koh ◽  
Ching Herny Chin ◽  
Yi Fei Tan ◽  
Tan Ching Ng
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Interarrival Time ◽  
Negative Customers ◽  
Queue Length Distribution ◽  
Alternative Approach ◽  
Stationary Queue Length ◽  
Stationary Queue Length Distribution ◽  
Stationary Probabilities

A single-server queueing system with negative customers is considered in this paper. One positive customer will be removed from the head of the queue if any negative customer is present. The distribution of the interarrival time for the positive customer is assumed to have a rate that tends to a constant as time t tends to infinity. An alternative approach will be proposed to derive a set of equations to find the stationary probabilities. The stationary probabilities will then be used to find the stationary queue length distribution. Numerical examples will be presented and compared to the results found using the analytical method and simulation procedure. The advantage of using the proposed alternative approach will be discussed in this paper.

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.

scholarly journals Alternative approach based on roots for computing the stationary queue-length distributions in GIX/M(1, b)/1 single working vacation queue

2020 ◽  
Author(s):  
Miaomiao Yu
Keyword(s):  
Queue Length ◽  
Transition Probability ◽  
Length Distribution ◽  
Transition Probability Matrix ◽  
Batch Arrival ◽  
Embedded Markov Chain ◽  
Working Vacation ◽  
Queue Length Distribution ◽  
Stationary Queue Length ◽  
Number Of Customers

The purpose of this paper is to present an alternative algorithm for computing the stationary queue-length and system-length distributions of a single working vacation queue with renewal input batch arrival and exponential holding times. Here we assume that a group of customers arrives into the system, and they are served in batches not exceeding a specific number b. Because of batch arrival, the transition probability matrix of the corresponding embedded Markov chain for the working vacation queue has no skip-free-to-the-right property. Without considering whether the transition probability matrix has a special block structure, through the calculation of roots of the associated characteristic equation of the generating function of queue-length distribution immediately before batch arrival, we suggest a procedure to obtain the steady-state distributions of the number of customers in the queue at different epochs. Furthermore, we present the analytic results for the sojourn time of an arbitrary customer in a batch by utilizing the queue-length distribution at the pre-arrival epoch. Finally, various examples are provided to show the applicability of the numerical algorithm.

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