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 ε-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(ε)) 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 η n (ε) when the queue size is n with n>N * . Moreover, for all n with n>N * , lim ε→ 0 η n (ε)=∞. On the other hand, N * and θ * are not functions of ε and they depend only on the parameters defining the system. Finally, a stationary and Markovian symmetric Nash equilibrium strategy does not exist.