Question

Customers arrive at a two-server service station according to a Poisson process with rate \(\lambda .\) Whenever a new customer arrives, any customer that is in the system immediately departs. A new arrival enters service first with server 1 and then with server 2\. If the service times at the servers are independent exponentials with respective rates \(\mu_{1}\) and \(\mu_{2}\), what proportion of entering customers completes their service with server 2?

Short answer

The proportion of customers completing their service with server 2 is given by the probability of no arrivals during the service time at server 1, which can be calculated as \(e^{-\lambda \frac{1}{\mu_1}}\).

Step-by-step solution

Analyze customer behavior and transitions

To start, we should consider a customer's possible outcomes when arriving at the service station: 1. The customer arrives and there is no other customer in the system, so they proceed to server 1, and then to server 2. 2. The customer arrives when another customer is in the system. In this case, the current customer immediately leaves the system, and the new arrival takes their place by starting service with server 1. Recall that we want to find the proportion of customers who complete their service at server 2. The only situation where a customer would fail to complete service with server 2 is if a new customer arrives while they are being served by server 1. If a customer can complete service with server 1 without being interrupted by another arrival, they will then move on to server 2 and complete their service.

Determine the probability that a customer is uninterrupted at server 1

We need to determine the probability that no customer arrives while a particular customer is being served by server 1. The service time for server 1 is exponentially distributed with rate \(\mu_1\), which means the average service time is \(\frac{1}{\mu_1}\). We want to find the probability that no arrival happens during the average service time of server 1. Since the arrival process follows a Poisson process with rate \(\lambda\), the probability of no arrival during an interval of length t is given by: \(P(\text{No arrivals in time interval }t) = e^{-\lambda t}\) In our case, \(t = \frac{1}{\mu_1}\). Plugging this into the formula, we have: \(P(\text{No arrivals during service time at server 1}) = e^{-\lambda \frac{1}{\mu_1}}\)

Obtain the proportion of customers completing service with server 2

The customers who complete service with server 2 are the ones who are not interrupted during their service time at server 1. Thus, the proportion of customers completing their service with server 2 is the probability of no arrivals during the service time at server 1: Proportion of customers completing service with server 2 = \(e^{-\lambda \frac{1}{\mu_1}}\)