Question

Potential customers arrive to a single-server hair salon according to a Poisson process with rate \(\lambda .\) A potential customer who finds the server free enters the system; a potential customer who finds the server busy goes away. Each potential customer is type \(i\) with probability \(p_{i}\), where \(p_{1}+p_{2}+p_{3}=1\). Type 1 customers have their hair washed by the server; type 2 customers have their hair cut by the server; and type 3 customers have their hair first washed and then cut by the server. The time that it takes the server to wash hair is exponentially distributed with rate \(\mu_{1}\), and the time that it takes the server to cut hair is exponentially distributed with rate \(\mu_{2}\). (a) Explain how this system can be analyzed with four states. (b) Give the equations whose solution yields the proportion of time the system is in each state. In terms of the solution of the equations of (b), find (c) the proportion of time the server is cutting hair; (d) the average arrival rate of entering customers.

Short answer

(a) The system can be analyzed with four states: - State 0: The system is empty (no customer is being served). - State 1: A customer of Type 1 is being served (hair is being washed). - State 2: A customer of Type 2 is being served (hair is being cut). - State 3: A customer of Type 3 is being served (hair is being washed and will be cut subsequently). (b) The balance equations for the four states are: - State 0: \(P_0 \lambda = (P_1 \mu_1) + (P_2 \mu_2) + (P_3 \mu_1)\). - State 1: \(P_0 (\lambda p_1) = P_1 \mu_1\). - State 2: \(P_0 (\lambda p_2) = P_2 \mu_2\). - State 3: \(P_0 (\lambda p_3) = P_3 \mu_1\). Additionally, \(P_0 + P_1 + P_2 + P_3 = 1\). (c) The proportion of time the server is cutting hair, \(P_{cut}\), is: \[P_{cut} = P_2 + \frac{P_3}{\mu_1 + \mu_2}.\] (d) The average arrival rate of entering customers, \(A_r\), is: \[A_r = \lambda P_0.\]

Step-by-step solution

a) Analyzing the System with Four States

To analyze the system with four states, we need to consider the possible occurrences in each state. We label the states as follows: - State 0: The system is empty (no customer is being served). - State 1: A customer of Type 1 is being served, i.e., their hair is being washed. - State 2: A customer of Type 2 is being served, i.e., their hair is being cut. - State 3: A customer of Type 3 is being served, i.e., their hair is being washed and will be cut subsequently. We describe a birth-death process with state transitions corresponding to arrivals and completions of service.

b) Finding the Equations for Proportion of Time

Now, let \(P_i\) denote the proportion of time the system is in state \(i\), where \(i=\{0,1,2,3\}\). We can write out the balance equations for the four states as follows. Keep in mind that the rate of arriving customers is \(\lambda\) and the service rates are \(\mu_1\) for hair washing and \(\mu_2\) for hair cutting: - State 0: \(P_0 \lambda = (P_1 \mu_1) + (P_2 \mu_2) + (P_3 \mu_1)\). - State 1: \(P_0 (\lambda p_1) = P_1 \mu_1\). - State 2: \(P_0 (\lambda p_2) = P_2 \mu_2\). - State 3: \(P_0 (\lambda p_3) = P_3 \mu_1\). Additionally, the sum of all probabilities must be equal to one, hence, \(P_0 + P_1 + P_2 + P_3 = 1\).

c) Proportion of Time Server is Cutting Hair

To find the proportion of time the server is cutting hair, we should sum up the proportion of time in State 2 (when a Type 2 customer is being served) and the proportion of time in State 3 (when a Type 3 customer is being served, but only after having their hair washed, i.e., the server is now cutting their hair). We can denote this by \(P_{cut}\): \[P_{cut} = P_2 + \frac{P_3}{\mu_1 + \mu_2}.\]

d) Average Arrival Rate of Entering Customers

To find the average arrival rate of entering customers, we will use the fact that the total rate of customers entering is \(\lambda\). However, only those customers who find the server free will enter the system. Thus, the average arrival rate of entering customers \(A_r\) can be calculated using the probability of finding the server idle (State 0): \[A_r = \lambda P_0.\]