Steady state results for the M/M(a,b)/c batch-service system

The transportation system considered in this paper has a number of vehicles with capacity constraint, which take passengers from a source terminal to various destinations and return to the terminal. The trip times are considered to be independent and identically distributed random variables with a common exponential distribution. Passengers arrive at the terminal in accordance with a Poisson process. The system is operated under the following policy: when a vehicle is available and there are at least ‘a’ passengers waiting for service, then a vehicle is dispatched immediately. A recursive algorithm is derived to obtain the steady-state probability P(m, j) that there are m idle vehicles and j waiting passengers in the queue. Analytical expressions have been derived for passenger queue length distribution, average passenger queue length, the r-th moment of passenger waiting time in the queue, service batch size distribution and the average service batch size, all in terms of P(0,0).