Question

Consider a single server queuing system where customers arrive according to a Poisson process with rate \(\lambda\), service times are exponential with rate \(\mu\), and customers are served in the order of their arrival. Suppose that a customer arrives and finds \(n-1\) others in the system. Let \(X\) denote the number in the system at the moment that customer departs. Find the probability mass function of \(X\). Hint: Relate this to a negative binomial random variable.

Short answer

The probability mass function of the number of customers, \(X\), in the system when the customer who arrived to find \(n-1\) others in the system departs can be expressed as a negative binomial distribution: \(P(X=k) = \binom{k+n-1}{k}\left(\frac{\mu}{\lambda + \mu}\right)^{n}\left(\frac{\lambda}{\lambda + \mu}\right)^{k}\) where \(k\) represents the number of arrivals during the service time of the given customer, \(\lambda\) is the arrival rate, and \(\mu\) is the service rate.

Step-by-step solution

Identify the Variables

In this queuing system, we have the arrival rate \(\lambda\), the service rate \(\mu\), and we are given that there are \(n-1\) customers already in the system when the customer of interest arrives. Let's use the negative binomial random variable \(X\) to represent the number of customers in the system when our customer departs.

Understand the Relationship with a Negative Binomial Distribution

A negative binomial distribution is the probability distribution of the number of failures before the \(r\)-th success in a Bernoulli process with probability \(p\). Here, we consider each new arrival during the service time of the chosen customer as a "failure" and the departure of the given customer as a "success." The given customer will be the \(n\)-th customer to be served: we already had \(n-1\) customers in the system, and we need to track the number of arrivals until this customer leaves. Therefore, we are interested in the number of failures before the \(n\)-th success, and we can model this using a negative binomial distribution.

Calculate the Probability of an Arrival

We need to find the probability \(p\) of an arrival during the service time of the given customer. Since the interarrival times follow an exponential distribution with rate \(\lambda\) and the service times follow an exponential distribution with rate \(\mu\), the probability that a new arrival occurs before the given customer's service time is: \(p = \frac{\lambda}{\lambda + \mu}\)

Apply the Negative Binomial Formula

We can now apply the negative binomial formula to our problem. The probability mass function of a negative binomial distribution with parameters \(r = n\) (number of successes we are interested in) and \(p\) (probability of an arrival) is \(P(X=k) = \binom{k+n-1}{k}(1-p)^{n}p^{k}\) where \(k\) represents the number of arrivals or "failures" during the given customer's service time.

Write the Probability Mass Function of X

Using the probability \(p = \frac{\lambda}{\lambda + \mu}\) from Step 3, and the Negative Binomial formula from Step 4, the probability mass function of \(X\) can be expressed as: \(P(X=k) = \binom{k+n-1}{k}\left(\frac{\mu}{\lambda + \mu}\right)^{n}\left(\frac{\lambda}{\lambda + \mu}\right)^{k}\) This expression represents the probability mass function of the number of customers in the system when the customer who arrived to find \(n-1\) others in the system departs.