Reneging from Processor Sharing Systems and Random Queues

[In a memoryless processor sharing system and in a memoryless random queue, both with some cost structure, it is shown that there exists an ∈-Nash equilibrium symmetric reneging strategy which is Markovian and stationary with respect to the queue length and which is of the following (N*, θ*, η(∈)) form: (i) stay in the system when the queue size is smaller than some specified critical level N*; (ii) randomly renege with constant specified rate θ* when the queue size is N*; (iii) randomly renege with large enough rate $\eta _{n}(\epsilon)$ when the queue size is n with n > N*. Moreover, for all n with n > N*, ${\rm lim}_{\epsilon \rightarrow 0}\ \eta _{n}(\epsilon)=\infty $. On the other hand, N* and θ* are not functions of ∈ and they depend only on the parameters defining the system. Finally, a stationary and Markovian symmetric Nash equilibrium strategy does not exist.]