Individually and socially optimal joining rules for an egalitarian processor-sharing queue under different information scenarios

• Optimal joining rules for processor-sharing queue under different information levels are considered.
• Employing left-multiplication transformation, the expected conditional sojourn time is obtained.
• Socially optimal threshold is no longer a lower bound for the individually optimal threshold.
• Equilibrium and socially optimal mixed strategies are derived in fully unobservable case.

This paper studies joining behavior of customers into an M/M/1M/M/1 egalitarian processor-sharing (PS) queue. By constructing a left-multiplication transformation and using its matrix representation, we obtain the expected conditional sojourn time of a tagged customer. Then, in the fully observable case, we first consider the joining strategy in a decentralized manner, that is, arriving customers observe the queue size and then decide whether or not to join the queue based on the net benefit they will obtain upon the completion of service. Secondly, we derive the threshold strategy that will yield the system’s maximal expected profit, to reach the so-called social welfare optimization. Finally, Nash equilibrium and socially optimal mixed strategies are derived in the fully unobservable case. Moreover, some numerical examples are provided to explore the impact of system parameters on customer’s joining behavior.