Numerical investigation of queuing models with delayed feedbacks

We consider queuing system with N, 1 < N < да, identical channels and Poisson input flow of primary calls (p-calls). If upon arrival of a p-call there is at least one free channel, then it accepted into the system and the service process immediately begins. Otherwise, i.e. if at this moment all the channels of the system are busy, then the incoming p-call, according to the Bernoulli scheme, either goes into orbit with probability a(r),0 < a(r) < 1, to repeat service after some random time, or leaves the system with an complementary probability 1 - a (r), where r means the current number of calls in orbit. The p-call service times are independent and identically distributed random variables that have an exponential distribution function with a common mean. After completing the service, the p-call, according to the Bernoulli scheme, either with probability а(n) join to orbit for repeat the service after a random time, or with an complementary probability 1 - a( n) the call leaves the system. Here, the parameter n indicates the number of occupied channels immediately before the moment the service is completed for this call, n = 1,2,..., N. Calls that require repeated service and calls that were not able to access the channel at the time of arrival, organize a common orbit of repeated calls (r-calls) with a maximum size R, 0 < R < ю. In the case R < ю, the call is accepted into orbit, if at the time of its arrival the total number of repeated calls in the orbit is less than R. Otherwise, he/she leaves the system with a probability one. The orbit generates service requests at random times that obey the exponential distribution with a finite mean. It is assumed that r-calls and p-calls are identical in duration of their service time. Repeated calls are persistent, i.e. if upon arrival of the r-call all the channels of the system are busy, then, according to the Bernoulli scheme, he/she leave the system with probability p(r), 0 < p(r) < 1, or he/she remain in orbit with an complementary probability 1 - p (r) , where r means the current number of r-calls in orbit, r = 1,2,...,R. It is shown that the mathematical model of this system is a certain two-dimensional Markov chain. An algorithm for constructing the generating matrix of the constructed chain is developed. The paper proposes exact and approximate methods for calculating the steady-stationary probabilities of the constructed two-dimensional Markov chain. Formulas are obtained for calculating the characteristics of the system - the probability of loss of primary and repeated calls, the average number of busy channels and the average number of repeated calls in orbit. The exact method is based on the application of the balance equation method for steady-state probabilities, and it is effective for moderate-dimensional models. An approximate method for studying large-dimensional models, which is based on the principles of state space merging of the multidimensional Markov chains, is developed as well. The proposed approaches also allow us to investigate models with a constant intensity of retrial calls (i.e., a model in which only the r-call at the head of the queue of repeated calls can generate a request) and with impatient r-calls in orbit. Moreover, in cases where the indicated above probabilities are constant values, it is possible to obtain explicit formulas for calculating the characteristics of models with infinite orbit size. The results of numerical experiments are demonstrated and their analysis is given.

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