Numerical investigation of queue with heteregeneous servers and randomized N-policy

We consider queueing system that contains two heterogeneous servers: fast server (F-server) and slow server (S-server). Input flow is Poisson one with rate λ where is independent on status of servers. Service times in both servers are independent random variables with exponential distributions; average service time in F-server and S-server are equal to μ _F^-1 и μ _S^-1 respectively, where μ_F > μ_S. It is assumed that the F-server remains awake at all times while the S-server switching on if queue length reaches some threshold N,N < ∞ . Switching scheme of the S-server is defined as follows. If upon arrival of call queue lenght is no less than N then the S-server is switching on with probability (w.p.) α,0 <α< 1, and with complementary probability 1-α this server remains in sleep mode (i.e. in status switch off). Upon completion of servicing of an call in the S-server, it selects one call from the queue for the service if the queue length at this moment is greater than the value N; otherwise, the S-server goes into sleep mode. In model with finite buffer an arrived call is blocked if at this moment buffer is fully occupied and the S-server is swithcing on; if upon arrival buffer is fully occupied and the S-server is in sleep mode then either the S-server is switching on w.p. a or with complementary probability an arrived call is blocked. Our problem is calculating the joint distribution of number of calls in the system and status of the S-server and developing the method to finding the performance measures of system. It is shown that mathematical model of the investigated systems are two-dimensional Markov chains (2D MC). The algorithms to constructing the generating matrix of these 2D MC are developed. The exact and approximate methods to calculating the steady-state probabilities of the indicated 2D MC are proposed. Explicit formulas to calculation of the following performance measures are developed: average number of calls in the system, average sojourn time in the system, intensity of switching on of the S-server, probability of blocking of calls in the system with finite buffer. The approximate method is based on principles of state space merging of 2D MC and it is designed to investigation of large scale models. It is shown that the developed approximate formulas have high accuracy. The problem of minimization of total cost via choosing the optimal value of threshold parameter N is solved. Results of numerical experiments are demonstrated and their analysis is performed.

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