OPTIMAL CONTROL POLICIES FOR AN M/M/1 QUEUE WITH A REMOVABLE SERVER AND DYNAMIC SERVICE RATES

We consider an M/M/1 queue with a removable server that dynamically chooses its service rate from a set of finitely many rates. If the server is off, the system must warm up for a random, exponentially distributed amount of time, before it can begin processing jobs. We show under the average cost criterion, that work conserving policies are optimal. We then demonstrate the optimal policy can be characterized by a threshold for turning on the server and the optimal service rate increases monotonically with the number in system. Finally, we present some numerical experiments to provide insights into the practicality of having both a removable server and service rate control.

A note on the waiting time in M[X]/G/1 queueing systems with a removable server

For the M[X]/G/1 queueing model with a general exhaustive-service vacation policy, it has been proved that the Laplace-Stieltjes transform (LST) of the steady-state distribution function of the waiting time of a customer arriving while the server is active is the product of the corresponding LST in the bulk arrival model with unremovable server and another LST. The expression given for the latter, however, is valid only under the assumption that the number of groups arriving in an inactive phase is independent of the sizes of the groups. We here give an expression which holds in the general case. For the N-policy case, we also give an expression for the LST of the steady-state distribution function of the waiting time of a customer arriving while the server is inactive.

A note on the waiting time in M[X]/G/1 queueing systems with a removable server

For the M[X]/G/1 queueing model with a general exhaustive-service vacation policy, it has been proved that the Laplace-Stieltjes transform (LST) of the steady-state distribution function of the waiting time of a customer arriving while the server is active is the product of the corresponding LST in the bulk arrival model with unremovable server and another LST. The expression given for the latter, however, is valid only under the assumption that the number of groups arriving in an inactive phase is independent of the sizes of the groups. We here give an expression which holds in the general case. For the N-policy case, we also give an expression for the LST of the steady-state distribution function of the waiting time of a customer arriving while the server is inactive.