Dequan Yue

Analysis of an M/M/c/N Queueing System with Balking, Reneging, and Synchronous Vacations

In this chapter, we present an analysis for an M/M/c/N queueing system with simultaneous balking, reneging, and synchronous vacations of servers. By using the blocked matrix method, we obtain the steady-state probability vector presented by the inverses of two matrices. The computing of the inverses of the two matrices is discussed. Then, we calculate the steady-state probabilities by using the elements of the inverses of the two matrices. We also derive the conditional stationary distribution of the queue length and waiting time.

A heterogeneous two-server network system with balking and a Bernoulli vacation schedule

In this paper, we study a two-server Markovian network system with balking and a Bernoulli schedule under a single vacation policy where servers have different service rates. After every service, only one server may take a vacation or continue to stay in the system. The vacation time follows an exponential distribution. An arriving customer finding both servers free will choose the faster server. If the customer finds both servers are not free, then this customer may join the system or balk. For this system, we obtain the steady state condition, the stationary distribution of the number of customers in the system and the mean system size by using a matrix geometric method. Some special cases are deduced, which match with earlier exiting results. Extensive numerical illustrations are provided. Motivation for this system model also comes from some computer communication networks with different types of traffic such as real-time traffic and non-real-time traffic, where messages can be processed by two channels (servers) with different transmission rates. The behavior of abandoning messages can be equated with the balking of customers in this system model.

Analysis of a Multi-server Queueing-inventory System with Non-homogeneous Poisson Arrivals

In this paper, we consider a multi-server queueing-inventory system with non-homogeneous Poisson arrivals, where the service times and lead times are exponentially distributed. The arrival process of the customers depends on the inventory level. When the on-hand inventory level decreases to zero, one item is replenished immediately with zero lead time. We formulated the system as a quasi-birth-and-death (QBD) process and obtain a stability condition of the system. Then, we compute the stationary distribution of the system and some main performance measures which are used to derive the total cost function. Finally, the computation of optimal (s, S) policy are illustrated by numerical examples.

Analysis of an M/M/c Queueing System with Impatient Customers and Synchronous Vacations

We consider an M/M/c queueing system with impatient customers and a synchronous vacation policy, where customer impatience is due to the servers’ vacation. Whenever a system becomes empty, all the servers take a vacation. If the system is still empty, when the vacation ends, all the servers take another vacation; otherwise, they return to serve the queue. We develop the balance equations for the steady-state probabilities and solve the equations by using the probability generating function method. We obtain explicit expressions of some important performance measures by means of the two indexes. Based on these, we obtain some results about limiting behavior for some performance measures. We derive closed-form expressions of some important performance measures for two special cases. Finally, some numerical results are also presented.