A class of multi-server queueing system with server failures
Introduction
The problem of queueing systems with server failures is of continuing interest to many researchers. Several models have been built and analyzed. White and Christie (1958) introduced queueing systems with server interruptions. They pointed out the similarity of queueing with breakdowns to queueing with preemptive priority and worked out two models of breakdown effects when they discussed the preemptive priority queueing system M/G/1 and found the steady-state solutions to it. Keilson and Kooharian (1962) discussed the M/G (M, G)/1 queue with interruptions by server breakdowns or the arrival of customers with higher priority. Avi-Itzhak and Naor (1962) also discussed M/G (M, G)/1 queueing system with server breakdowns. Neuts and Lucantoni (1979) discussed the model of M/M (M, M)/N queue with server failures. Dantzer, Mitrani, and Robert (2001) discussed the asymptotic behavior of the M/M/n queue, with servers subject to independent breakdowns and repairs, in the limit where the number of servers tends to infinity and the repair rate tends to 0, such that their product remains finite. They also analyzed the behavior of the M/M/M/M/∞ in heavy traffic when the traffic intensity approaches 1. Federgruen and Green (1986) discussed an M/G (G, G)/1 queueing system with server interruptions. Bounds and approximations for the mean waiting time, probability of delay and steady-state distribution of the number in system were derived. These results were exact for the case of an M/G (G, M)/1 queueing system. Federgruen and Green (1988) presented an exact solution method for an M/G (PH, G)/1 queueing system with server breakdowns. Zhang, Vickson, and van Eenige (1997) treated two-threshold policies for an M/G/1 queue with two types of generally distributed random vacations: type 1 (long) and type 2 (short) vacations, studied an M/G/1 queue with an exceptional first vacation in Zhang and Ernest (1998), and considered a single server queueing system with Poisson arrivals and multiple vacation types in Zhang et al. (2001).
Where queueing systems with unreliable servers are concerned, most research that has been done focuses on one-server systems. When multi-server is studied the work has been limited to systems with Poisson arrival process and exponential service time. However, in many situations we need to consider non-exponential service times.
The system we study is such that the service times have phase type distributions and the number of available repair people does not exceed the number of servers. We let the time between failures and repair times have exponential distributions. If at breakdown of a server all repair persons are busy then the failed server has to wait until a repair person is available. The GI/PH/n queueing system (for which the M/PH/n is a special case) without server breakdowns was discussed in Neuts (1981, p. 206).
Section snippets
Notation and assumptions
We use the following notations throughout this paper:
- λ
intensity or rate of the Poisson arrival process
- θ
rate of the exponential distribution of the server inter-failure time
- γ
rate of the exponential distribution of the server repair time
- L
mean queue length
- W
mean of the waiting time
- n
number of servers
- k
number of repair persons (1 ⩽ k ⩽ n)
- (β, S)
representation of the phase type distribution of the service time
- μ
expected service rate
- c
capacity of the queueing system
- π
limiting or equilibrium distribution of queue
Modeling the infinite buffer system
We consider the M/PH/n queue with server failures. The representation of the phase type distributed service time is (β, S) of order v, which means the number of phases of a server is v and S0 = 0 − Se, where 0 is a column vector of all entries 0 and e is a column vector of all entries 1. So, the state space of the system is E = {(i, j, m1, m2, … , mn);i ⩾ 0, 0 ⩽ j ⩽ n, 1 ⩽ mp ⩽ v, p = 1, 2, …, n}, where i denotes the number of customers in the system, j denotes the number of available servers, and mp denotes the phase of the
The case of finite buffer
Now we consider the case of finite buffer and show an application of this research in mobile wireless communication. A finite state-space queueing system in which there are n identical unreliable servers with phase type distributed service times and k repair persons is considered. Computer programs are developed to compute the state probabilities and performance measurements.
Conclusions
This paper discussed the modeling, stability condition analysis and stationary distribution calculation of the M/PH/n queueing system, and the modeling, stationary distribution calculation of the M/PH/n/c queueing system. Application example was also provided to demonstrate the application of the research.
Acknowledgements
The authors thank the referees and editor for their critical reading of the manuscript and useful suggestions and comments that have significantly improved the quality of the paper.
References (12)
- et al.
Optimal two-threshold policies in an M/G/1 queue with two vacation types
Performance Evaluation
(1997) - et al.
Some problems with the service station subject to breakdown
Operations Research
(1962) - et al.
Large scale and heavy traffic asymptotics for systems with unreliable servers
Queueing Systems
(2001) - et al.
Queueing systems with service interruptions
Operations Research
(1986) - et al.
Queueing systems with service interruptions II
Naval Research Logistics Quarterly
(1988) - et al.
Finite birth-and-death models in randomly changing environments
Advances in Applied Probability
(1984)