A BMAP/PH/N SYSTEM WITH IMPATIENT REPEATED CALLS

2007 ◽  
Vol 24 (03) ◽  
pp. 293-312 ◽  
Author(s):  
VALENTINA I. KLIMENOK ◽  
DMITRY S. ORLOVSKY ◽  
ALEXANDER N. DUDIN
Keyword(s):  
Markov Chain ◽  
Time Distribution ◽  
Arrival Process ◽  
Service Time Distribution ◽  
State Distribution ◽  
Unsuccessful Attempt ◽  
Batch Markovian Arrival Process ◽  
Repeated Calls ◽  
Phase Type ◽  
Multi Server

A multi-server queueing model with a Batch Markovian Arrival Process, phase-type service time distribution and impatient repeated customers is analyzed. After any unsuccessful attempt, the repeated customer leaves the system with the fixed probability. The behavior of the system is described in terms of continuous time multi-dimensional Markov chain. Stability condition and an algorithm for calculating the stationary state distribution of this Markov chain are obtained. Main performance measures of the system are calculated. Numerical results are presented.

scholarly journals Analysis of a Priority Queue with Phase-Type Service and Failures

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Alexander Dudin ◽  
Sergei Dudin
Keyword(s):  
Markov Chain ◽  
Performance Measures ◽  
Priority Queue ◽  
Arrival Process ◽  
Single Server ◽  
Service Time Distribution ◽  
New Class ◽  
Phase Type ◽  
Chain Problem ◽  
Insight Into

We consider a single server queue with two types of customers. We propose a discipline of flexible priority in access that combines the features of randomization and the threshold type control. We introduce a new class of distributions, phase-type with failures (PHF) distribution, that generalizes the well-known phase-type (PH) distribution to the case when failures can occur during service of a customer. The arrival flow is described by the marked Markovian arrival process. The service time distribution is of PHF type with the parameters depending on the type of a customer. Customers of both types can be impatient. Behavior of the system is described by the multidimensional Markov chain. Problem of existence and computation of the stationary distribution of this Markov chain is discussed in brief as well as the problem of computation of the key performance measures of the system. Numerical examples are presented that give some insight into behavior of the system performance measures under different values of the parameters defining the strategy of customers access to service.

scholarly journals Analysis of an MMAP/PH1, PH2/N/∞ queueing system operating in a random environment

2014 ◽  
Vol 24 (3) ◽  
pp. 485-501 ◽  
Author(s):  
Chesoong Kim ◽  
Alexander Dudin ◽  
Sergey Dudin ◽  
Olga Dudina
Keyword(s):  
Random Environment ◽  
Queueing System ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Contact Center ◽  
Phase Type Distribution ◽  
Phase Type ◽  
Multi Server

Abstract A multi-server queueing system with two types of customers and an infinite buffer operating in a random environment as a model of a contact center is investigated. The arrival flow of customers is described by a marked Markovian arrival process. Type 1 customers have a non-preemptive priority over type 2 customers and can leave the buffer due to a lack of service. The service times of different type customers have a phase-type distribution with different parameters. To facilitate the investigation of the system we use a generalized phase-type service time distribution. The criterion of ergodicity for a multi-dimensional Markov chain describing the behavior of the system and the algorithm for computation of its steady-state distribution are outlined. Some key performance measures are calculated. The Laplace-Stieltjes transforms of the sojourn and waiting time distributions of priority and non-priority customers are derived. A numerical example illustrating the importance of taking into account the correlation in the arrival process is presented

Characterization for the queueing system M/G/∞

1973 ◽  
Vol 74 (1) ◽  
pp. 141-143 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Distribution Function ◽  
Time Distribution ◽  
Queueing System ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Traffic Intensity ◽  
Infinitely Divisible ◽  
Waiting Room ◽  
Room Size ◽  
Arrival Intensity

Consider a queueing system M/G/s with the arrival intensity λ, the service time distribution function B(t) (B(0) < 1) having a finite mean and the waiting room size N ≤ ∞. If s < ∞ and N = ∞, we shall also assume that its relative traffic intensity is less than 1. Since the arrival process of this system is Poisson, it is immediate that in this case the distribution of the number of arrivals during an interval is infinitely divisible.

The multiclass GI/PH/N queue in the Halfin-Whitt regime

2000 ◽  
Vol 32 (02) ◽  
pp. 564-595 ◽  
Author(s):  
A. A. Puhalskii ◽  
M. I. Reiman
Keyword(s):  
Queue Length ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Service Time Distribution ◽  
Time Process ◽  
Dimensional Diffusion ◽  
Phase Type ◽  
Waiting Time Process ◽  
Multiple Customer Classes ◽  
Limit Diffusion

We consider a multiserver queue in the heavy-traffic regime introduced and studied by Halfin and Whitt who investigated the case of a single customer class with exponentially distributed service times. Our purpose is to extend their analysis to a system with multiple customer classes, priorities, and phase-type service distributions. We prove a weak convergence limit theorem showing that a properly defined and normalized queue length process converges to a particular K-dimensional diffusion process, where K is the number of phases in the service time distribution. We also show that a properly normalized waiting time process converges to a simple functional of the limit diffusion for the queue length.

Waiting-Time Percentiles in the Multi-server Mx/G/c Queue with Batch Arrivals

1987 ◽  
Vol 1 (1) ◽  
pp. 75-96 ◽  
Author(s):  
A. M. Eikeboom ◽  
H. C. Tijms
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Linear Interpolation ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Batch Arrivals ◽  
Special Cases ◽  
Moment Approximation ◽  
Multi Server ◽  
Validation Experiments

This paper deals with the MX/G/c queue. Using analytical results for the special cases of the MX/M/c queue and the MX/D/c queue, a two-moment approximation is proposed for the waiting-time percentiles in the general case. This approximation is based on a linear interpolation with respect to the squared coefficient of variation of the service time distribution. Validation experiments indicate that this approximation performs quite well for practical purposes. In particular, the practically important percentiles in the tail of the waiting-time distribution are approximated extremely well.

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