Abstract
A novel multi-server vacation queuing model is considered. The distinguishing feature of the model, compared to the standard queues, is the self-sufficiency of servers. A server can terminate service and go on vacation independently of the system manager and the overall situation in the system. The system manager can make decisions whether to allow the server to start work after vacation completion and when to try returning some server from a vacation to process customers. The arrival flow is defined by a general batch Markov arrival process. The problem of optimal choice of the total number of servers and the thresholds defining decisions of the manager arises. To solve this problem, the behavior of the system is described by the three-dimensional Markov chain with the special block structure of the generator. Conditions for the ergodicity of this chain are derived, the problem of computation of the steady-state distribution of the chain is discussed. Expressions for the key performance indicators of the system in terms of the distribution of the chain states are derived. An illustrative numerical result is presented.
Highlights
Queuing theory is one of the most quickly developing branches of applied probability due to the intensive appearance of new queuing models of real-world systems, telecommunication networks, in particular
The most common assumption made in the literature about multi-server queues is that there exists a finite or infinite pool of servers that provide service to arriving customers under the control of the system manager
A detailed description of the batch Markov arrival process (BMAP) can be found, e.g., in [37,38,39]; we present only very dense information about the BMAP that is necessary for the goals of this paper
Summary
Queuing theory is one of the most quickly developing branches of applied probability due to the intensive appearance of new queuing models of real-world systems, telecommunication networks, in particular. Service and other rates are changed at the epochs of the change of the state of the random environment These systems can be regarded as the generalization of unreliable queues to the cases of more flexible mechanisms of server activation and deactivation as well as partial reliability of the servers. Such systems are called in the literature as queuing-inventory systems, assembly-like systems, double-sided or queues with paired customers These systems are non-conservative because the servers can stay idle in the presence of a non-empty buffer if some additional resource required for service is currently unavailable. Customers may leave the queue without service (if the products are perishable or must be delivered to the consumer by a certain time), which may provoke substantial losses for the company If this number is too large, problems with providing enough work to the servers can arise.
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