Question

Arrivals to a three-server system are according to a Poisson process with rate \(\lambda\). Arrivals finding server 1 free enter service with \(1 .\) Arrivals finding 1 busy but 2 free enter service with \(2 .\) Arrivals finding both 1 and 2 busy do not join the system. After completion of service at either 1 or 2 the customer will then either go to server 3 if 3 is free or depart the system if 3 is busy. After service at 3 customers depart the system. The service times at \(i\) are exponential with rate \(\mu_{i}, i=1,2,3\). (a) Define states to analyze the above system. (b) Give the balance equations. (c) In terms of the solution of the balance equations, what is the average time that an entering customer spends in the system? (d) Find the probability that a customer who arrives when the system is empty is served by server 3 .

Short answer

To analyze the three-server system, we use a Continuous-Time Markov Chain model and define the states as follows: State 0 - All servers are free; State 1 - Server 1 is busy, but servers 2 and 3 are free; State 2 - Servers 1 and 2 are busy, but server 3 is free; State 3 - Server 3 is busy. We derive the balance equations and obtain the steady-state probabilities for each state. Using Little's law and these probabilities, we compute the average time that an entering customer spends in the system, W. Finally, we calculate the probability of a customer being served by server 3 as the ratio of the steady-state probability of state 3 to the sum of the probabilities of states 1, 2, and 3.

Step-by-step solution

Define States

Let's define the states of the system as follows: - State 0: All servers are free. - State 1: Server 1 is busy, but servers 2 and 3 are free. - State 2: Servers 1 and 2 are busy, but server 3 is free. - State 3: Server 3 is busy.

Give the Balance Equations

Now, we will write the balance equations for each state in terms of the steady-state probabilities P_i for state i=0,1,2,3. For state 0, \[\lambda P_0 = \mu_1 P_1\] For state 1, \[\lambda P_1 + \mu_3 P_3 = (\lambda + \mu_1) P_1\] For state 2, \[\lambda P_0 + \mu_2 P_2 = (\mu_1 + \mu_2) P_2\] For state 3, \[\mu_1 P_1 + \mu_1 P_2 = (\mu_3) P_3\]

Average Time in the System

We will now use the balance equations to find the average time a customer spends in the system. To do this, we first need to find the steady-state probabilities for each state, using the balance equations from Step 2. These probabilities are: \begin{equation} P_0 = \frac{\lambda}{\mu_1} P_1 \end{equation} \begin{equation} P_2 = \frac{\mu_1 + \mu_2}{\lambda} P_1 \end{equation} \begin{equation} P_3 = \frac{\mu_1}{\mu_3} P_1 + \frac{\mu_1}{\mu_3} P_2 \end{equation} Next, we can compute the average time a customer spends in the system using Little's law as follows: \begin{equation} L = \lambda W \end{equation} where L is the average number of customers in the system and W is the average time that an entering customer spends in the system. Using the steady-state probabilities, we can compute the average number of customers in the system, L: \begin{equation} L = P_1 + 2P_2 + 3P_3 \end{equation} Substituting the probabilities from equations (1), (2), and (3), as well as using equation (4), we can find the average time, W, that an entering customer spends in the system.

Probability of a Customer Served by Server 3

To find the probability of a customer being served by server 3, we can use the fact that customers only enter service with server 3 after they have completed service at either server 1 or server 2. Therefore, we can write this probability as follows: \begin{equation} P(\text{customer served by server 3}) = \frac{P_3}{P_1 + P_2 + P_3} \end{equation} Now we can plug in the steady-state probabilities of states 1, 2, and 3 (from equations 1, 2, and 3) to find the probability that a customer who arrives when the system is empty is served by server 3.