A small hair dresser shop operates with three hair stylists. It has two waiting seats so...

 

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A small hair dresser shop operates with three hair stylists. It has two waiting seats so that when new customers arrive and all stylists are busy and there is no vacant seat, they leave. The state space S of this "birth-and-death" process is S={0,1,2,3,4,5). where S indicates the number of customers either waiting or having the ongoing service. When 1 k 3 services are going on, the exponentially distributed time for one service to be completed is P=2.k. This process has with birth and death rates as follows: birth rates: 4 per hour, for k 0,1,..., 4, and A=0 for k5. death rates: 0-0, -2, 24, 36, and 0 for k>5. 1. Derive the stationary distribution for a number of customers in the shop. 2. Determine the limiting probability of the idle state, z=0, that is: lim PIX(t)=0) 3. Determine the limiting probability of losing customers because the shop is full, that is: lim P(X(t)=5] A small hair dresser shop operates with three hair stylists. It has two waiting seats so that when new customers arrive and all stylists are busy and there is no vacant seat, they leave. The state space S of this "birth-and-death" process is S={0,1,2,3,4,5). where S indicates the number of customers either waiting or having the ongoing service. When 1 k 3 services are going on, the exponentially distributed time for one service to be completed is P=2.k. This process has with birth and death rates as follows: birth rates: 4 per hour, for k 0,1,..., 4, and A=0 for k5. death rates: 0-0, -2, 24, 36, and 0 for k>5. 1. Derive the stationary distribution for a number of customers in the shop. 2. Determine the limiting probability of the idle state, z=0, that is: lim PIX(t)=0) 3. Determine the limiting probability of losing customers because the shop is full, that is: lim P(X(t)=5]

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To solve the given birthanddeath process we will use the balance equations and the concept of the st... View the full answer