Queuing systems with heterogeneous servers and state-dependent jump priorities informatics and programming

We consider the Markovian models of queuing systems with heterogeneous servers, calls of different types and jump priorities. Both types of models with finite individual buffers and an infinite common buffer are investigated. The calls of high priority are served by the fast server while the calls of low priority can be served in both the servers. State-dependent jump priorities determine the rules for transferring a low priority calls to a queue of high priority calls. In order to define state-dependent jump priorities two threshold parameters rl and rh are introduced as follows. If upon the arrival of low priority call the number of such kind of calls in system is more than rl and the number of high priority calls is less than rh, then in accordance to the Bernoulli trials either one of low priority call joins the queue of high priority calls or arrived call joins the end of low priority calls. If upon the arrival of low priority call the number of such kind of calls in system is more than rl and the number of high priority calls is more than rh, then the arrived call joins the end of low priority calls if there is a free place in buffer; otherwise it is a loss with the probability one. If upon the arrival of low priority call the number of appropriate buffers is full and the number of high priority calls is less than rh, then the low priotity call at the head of the own queue in accordance to the Bernoulli trials either joins the queue of high priority calls or the arrived call is lost. It is shown that the mathematical models of the investigated systems are a certain two-dimensional Markov chain with a finite or infinite set of states. The main performance measures are the loss probabilities of various kinds of calls (for the model with separate and finite buffers), the average rate of jumps from one queue to another, the average number of calls in the system and average waiting time in queues. The exact and approximate algorithms for calculating the steady-state probabilities as well as the performance measures are developed and the optimization problem is solved.

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