Imagine an $M/M/1$ queue wherein there is an option of either admitting an arrival, in which case there is a reward of Rupees $r$, but, a holding cost of Rupees $h$ for each time unit the admitted customer spends in the system is incurred or just rejecting the arrival. It is known that for the discounted reward structure, threshold policies are optimal: it is optimal to admit an arrival if and only if the number in the system is not more than a suitable $R^*$ which means that at optimality only a fraction of customers are admitted which can be viewed as a Quality of Service, QoS. Then, suppose that the Poisson arrival rate of the queue, $\lambda$, is a function of QoS; the QoS in turn depends on the arrival rate. Under mild conditions we first argue that if there is no equilibrium in such a queue, then an equilibrium set exists. We identify, such as multiple optimal threshold policies and different measures of QoS that can lead to the emergence of equilibrium sets. We illustrate the above with numerical examples that also bring out the role of some symbolic computation tools in this context. We also indicate some typos in the known algorithm to compute the optimal thresholds, which may be of independent interest. (Based on joint work with Kishor Patil and Sandhya Tripathi)
N. Hemachandra is a Professor of Industrial Engineering and Operations Research at IIT Bombay. His current academic interests include various operations research methodologies such as Markov decision models, queueing models and game theory, as well as their applications to problems arising from communication networks, supply chains, and power systems.