Comparing single-server loss systems in random environment

Considers a single-server loss system in a random environment. The environment is determined by a finite-state Markov process. When the environment is in state i, the arrival process is Poisson with rate λ_i, and the service time is exponential with rate μ _i=1,···m. We show that the blocking probability of this system is bounded from below (above) by that of the same system with a more (less) regular arrival and service pattern. This result supports Ross´s conjecture (1978) that the blocking probability is smaller when the arrival process is more regular and suggests its validity in scenarios with dependent arrival and service processes. Such structural properties are useful in obtaining bounds and approximations for system performance. We then fix the marginal service (arrival) process and search for an arrival (service) process with the same long-run rate that minimizes the blocking probability. It is shown that the optimal solution is the one such that the arrival and service rates are proportional. This result is in contrary with the case of independent arrival and service processes, where the system performance reaches its minimum when the arrival is replaced by Poisson, and it provides insight into the understanding of the effect of nonstationarity on system performance