scholarly journals On the GI/M/1 Queue with Vacations and Multiple Service Phases

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Jianjun Li ◽  
Liwei Liu
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Solution Method ◽  
System Size ◽  
Mean Waiting Time ◽  
The Matrix ◽  
Matrix Geometric Solution ◽  
Semi Markov Process ◽  
Stationary Waiting Time ◽  
Number Of Customers

This paper considers a GI/M/1 queue with vacations and multiple service phases. Whenever the system becomes empty, the server takes a vacation, causing the system to move to vacation phase 0. If the server returns from a vacation to find no customer waiting, another vacation begins. Otherwise, the system jumps from phase 0 to some service phase i with probability qi,  i=1,2,…,N. Using the matrix geometric solution method and semi-Markov process, we obtain the distributions of the stationary system size at both arrival and arbitrary epochs. The distribution of the stationary waiting time of an arbitrary customer is also derived. In addition, we present some performance measures such as mean waiting time of an arbitrary customer, mean length of the type-i cycle, and mean number of customers in the system at the end of phase 0. Finally, some numerical examples are presented.

scholarly journals The Geo/Geo/1+1 Queueing System with Negative Customers

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Zhanyou Ma ◽  
Yalin Guo ◽  
Pengcheng Wang ◽  
Yumei Hou
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
System Performance ◽  
Queue Length ◽  
Solution Method ◽  
Queueing System ◽  
Negative Customers ◽  
Idle Period ◽  
Matrix Geometric Solution ◽  
Qbd Process

We study a Geo/Geo/1+1 queueing system with geometrical arrivals of both positive and negative customers in which killing strategies considered are removal of customers at the head (RCH) and removal of customers at the end (RCE). Using quasi-birth-death (QBD) process and matrix-geometric solution method, we obtain the stationary distribution of the queue length, the average waiting time of a new arrival customer, and the probabilities of servers in busy or idle period, respectively. Finally, we analyze the effect of some related parameters on the system performance measures.

scholarly journals Neutrosophic Event-Based Queueing Model

2020 ◽  
pp. 48-55
Author(s):  
Mohamed Bisher Zeina ◽  
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Communication Systems ◽  
Queueing Systems ◽  
Real Life ◽  
Main Tool ◽  
Expected Number ◽  
Mean Waiting Time ◽  
Event Based ◽  
Number Of Customers

In this paper we have defined the concept of neutrosophic queueing systems and defined its neutrosophic performance measures. An important application of neutrosophic logic in queueing systems we face in real life were discussed, that is the neutrosophic events accuring times, because of its wide applications in networking and simulating communication systems specialy when probability distribution is not known, and because it’s more realistic to consider and to not ignore the imprecise events times. Event-based table of a neutrosophic queueing system was presented and its neutrosophic performance measures were derived, i.e. neutrosophic mean waiting time in queue, neutrosophic mean waiting time in system, neutrosophic expected number of customers in queue and neutrosophic expected number of customers in system. Neutrosophic Little’s Formulas (NLF) were also defined which is a main tool in queueing systems problems to make it easier finding performance measures from each other.

A single server Markovian queuing system with limited buffer and reverse balking

2021 ◽  
Vol 12 (7) ◽  
pp. 1774-1784
Author(s):  
Girin Saikia ◽  
Amit Choudhury
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Mutual Fund ◽  
System Size ◽  
Queuing System ◽  
Single Server ◽  
Insurance Company ◽  
Number Of Customers ◽  
Steady State Probabilities

The phenomena are balking can be said to have been observed when a customer who has arrived into queuing system decides not to join it. Reverse balking is a particular type of balking wherein the probability that a customer will balk goes down as the system size goes up and vice versa. Such behavior can be observed in investment firms (insurance company, Mutual Fund Company, banks etc.). As the number of customers in the firm goes up, it creates trust among potential investors. Fewer customers would like to balk as the number of customers goes up. In this paper, we develop an M/M/1/k queuing system with reverse balking. The steady-state probabilities of the model are obtained and closed forms of expression of a number of performance measures are derived.

Stationary waiting-time distributions in the GI/PH/1 queue

1981 ◽  
Vol 18 (4) ◽  
pp. 901-912 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Waiting Time ◽  
Embedded Markov Chain ◽  
Asymptotic Results ◽  
Waiting Time Distributions ◽  
Structured Systems ◽  
Queue Discipline ◽  
The Matrix ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Stationary Waiting Time

It is known that the stable GI/PH/1 queue has an embedded Markov chain whose invariant probability vector is matrix-geometric with a rate matrix R. In terms of the matrix R, the stationary waiting-time distributions at arrivals, at an arbitrary time point and until the customer's departure may be evaluated by solving finite, highly structured systems of linear differential equations with constant coefficients. Asymptotic results, useful in truncating the computations, are also obtained. The queue discipline is first-come, first-served.

scholarly journals Waiting Time Problems for Patterns in a Sequence of Multi-State Trials

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1893
Author(s):  
Bara Kim ◽  
Jeongsim Kim ◽  
Jerim Kim
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Probability Generating Function ◽  
Finite Collection ◽  
Numerical Inversion ◽  
Analytic Method ◽  
Waiting Time Distribution ◽  
Mean Waiting Time ◽  
The Matrix ◽  
The Mean

In this paper, we investigate waiting time problems for a finite collection of patterns in a sequence of independent multi-state trials. By constructing a finite GI/M/1-type Markov chain with a disaster and then using the matrix analytic method, we can obtain the probability generating function of the waiting time. From this, we can obtain the stopping probabilities and the mean waiting time, but it also enables us to compute the waiting time distribution by a numerical inversion.

On a property of the variance of the waiting time of a queue

1968 ◽  
Vol 5 (3) ◽  
pp. 702-703 ◽  
Author(s):  
D. G. Tambouratzis
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Mean Waiting Time ◽  
Queue Discipline ◽  
The Mean ◽  
Number Of Customers

In this note, we consider a queueing system under any discipline which does not affect the distribution of the number of customers in the queue at any time. We shall show that the variance of the waiting time is a maximum when the queue discipline is “last come, first served”. This result complements that of Kingman [1] who showed that, under the same assumptions, the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in the order of their arrival.

Waiting Time in M/G/1 Queues with Impolite Arrival Disciplines

1995 ◽  
Vol 9 (2) ◽  
pp. 255-267 ◽  
Author(s):  
Süleyman Òzekici ◽  
Jingwen Li ◽  
Fee Seng Chou
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Time Distribution ◽  
Iterative Procedure ◽  
Queueing System ◽  
Waiting Time Distribution ◽  
Special Model ◽  
Average Waiting Time ◽  
Random Position ◽  
Number Of Customers

We consider a queueing system where arriving customers join the queue at some random position. This constitutes an impolite arrival discipline because customers do not necessarily go to the end of the line upon arrival. Although mean performance measures like the average waiting time and average number of customers in the queue are the same for all such disciplines, we show that the variance of the waiting time increases as the arrival discipline becomes more impolite, in the sense that a customer is more likely to choose a position closer to the server. For the M/G/1 model, we also provide an iterative procedure for computing the moments of the waiting time distribution. Explicit computational formulas are derived for an interesting special model where a customer joins the queue either at the head or at the end of the line.

Generalised birth and death queueing processes: recent results

1977 ◽  
Vol 9 (1) ◽  
pp. 125-140 ◽  
Author(s):  
B. W. Conolly ◽  
J. Chan
Keyword(s):  
Waiting Time ◽  
Queueing Systems ◽  
System Size ◽  
Single Server ◽  
Traffic Intensity ◽  
Stable Regime ◽  
Mean Waiting Time ◽  
Service Rates ◽  
Queueing Processes ◽  
Effective Service

The systems considered are single-server, though the theory has wider application to models of adaptive queueing systems. Arrival and service mechanisms are governed by state (n)-dependent mean arrival and service rates λn and µn. It is assumed that the choice of λn and µn leads to a stable regime. Formulae are sought that provide easy means of computing statistics of effectiveness of systems. A measure of traffic intensity is first defined in terms of ‘effective’ service time and inter-arrival intervals. It is shown that the latter have a renewal type connection with appropriately defined mean effective arrival and service rates λ∗ and µ∗ and that in consequence the ratio λ∗/µ∗ is the traffic intensity, equal moreover to where is the stable probability of an empty system, consistent with other systems. It is also shown that for first come, first served discipline the equivalent of Little's formula holds, where and are the mean waiting time of an arrival and mean system size at an arbitrary epoch. In addition it appears that stable regime output intervals are statistically identical with effective inter-arrival intervals. Symmetrical moment formulae of arbitrary order are derived algebraically for effective inter-arrival and service intervals, for waiting time, for busy period and for output.

The analysis of a discrete time finite-buffer queue with working vacations under Markovian arrival process and PH-service time

2020 ◽  
Vol 54 (3) ◽  
pp. 675-691
Author(s):  
Qingqing Ye ◽  
Liwei Liu ◽  
Tao Jiang ◽  
Baoxian Chang
Keyword(s):  
Performance Measures ◽  
Discrete Time ◽  
Loss Probability ◽  
Combination Method ◽  
Arrival Process ◽  
Finite Buffer ◽  
The Matrix ◽  
Working Vacations ◽  
The Impact ◽  
Number Of Customers

In this paper, we study the discrete-time MAP/PH/1 queue with multiple working vacations and finite buffer N. Using the Matrix-Geometric Combination method, we obtain the stationary probability vectors of this model, which can be expressed as a linear combination of two matrix-geometric vectors. Furthermore, we obtain some performance measures including the loss probability and give the limit of loss probability as finite buffer N goes to infinite. Waiting time distribution is derived by using the absorbing Markov chain. Moreover, we obtain the number of customers served in the busy period. At last, some numerical examples are presented to verify the results we obtained and show the impact of parameter N on performance measures.

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