Abstract
We consider a queuing network with single-server nodes and heterogeneous customers. The number of customers, which can obtain service simultaneously, is restricted. Customers that cannot be admitted to the network upon arrival make repeated attempts to obtain service. The service time at the nodes is exponentially distributed. After service completion at a node, the serviced customer can transit to another node or leave the network forever. The main features of the model are the mutual dependence of processes of customer arrivals and retrials and the impatience and non-persistence of customers. Dynamics of the network are described by a multidimensional Markov chain with infinite state space, state inhomogeneous behavior and special structure of the infinitesimal generator. The explicit form of the generator is derived. An effective algorithm for computing the stationary distribution of this chain is recommended. The expressions for computation of the key performance measures of the network are given. Numerical results illustrating the importance of the account of the mentioned features of the model are presented. The model can be useful for capacity planning, performance evaluation and optimization of various wireless telecommunication networks, transportation and manufacturing systems.
Highlights
We deal with a queuing network which belongs to a relatively new category of semi-open queuing networks that recently were applied for the analysis of various real-world systems
An arriving primary customer is admitted to the network and starts processing only if the current number of customers in the network is less than the network capacity
The arrivals of primary customers and retrials of customers from the orbit depend on the same underlying process
Summary
The theory of queuing networks has a wide range of applications for modeling various real-world systems including telecommunication and logistic networks, health care, public transportation, production and manufacturing systems, see, for example, References [1,2,3,4] and so forth. We consider more general than the MAP-marked Markovian arrival process (MMAP), see, for example, Reference [24] This flow is heterogeneous and has several types of customers. Impatience of customers, that is, a possibility of abandonment during the waiting time after some period of waiting and non-persistence of customers staying in the orbit, that is, a possibility to renege from the orbit after any unsuccessful retrial, are typical for many real-world systems and networks They should be taken into account during performance evaluation and capacity planning. Reference [14], the model from Reference [10] is generalized to the case when there is no input buffer in the network and a customer arriving to the network when N customers receive service moves to the orbit and makes the retrials to obtain service.
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