Hystertic optimal policy of servers in front room and back room with switching time and switching cost

Many retail service are concerned with queueing theory problems, particularly ones which involve optimization. When a worker is being switched from one room to another, he has to move. In some cases this distance is very small, so it can be neglected. In the front room, workers serve customers who may be delayed in a queue. When the back room is interruptible, allowing a worker in the back room to switch to front room work with little delay or lost productivity. To optimize the operations of service facilities, we consider the amount of time that the workers are losing during the switch between the different types of work. After the completion of front room service if the customer is dissatisfied with its service for certain reason or if it received unsuccessful service, the customer may immediately join the second optional service and may leave the queue. The model is derived for switching from the front room to the back room, back room to the front room and loss of productivity due to switches between the rooms as switching cost. Our aim is to minimize customer waiting time with respect to switching servers from the front room to the back room.