Question

At the bank near where I work, there is a central queue served by four assistants. From past experience, I know that the service times of the four assistants are exponentially distributed with means 2 minutes, 3 minutes, 5 minutes and 6 minutes. One day Anita enters the bank to discover that all four assistants are busy, but no one is waiting to be served. (a) Find the distribution of the time that Anita will have to wait before she can move forward for service. Show that her expected waiting time is 50 seconds. (b) Calculate the probability that Anita will have to wait more than three minutes until an assistant is free to serve her. (c) Two minutes after entering the bank, Anita is still waiting. Another customer, Elias, enters the bank and stands behind her in the queue. (So now the same four people are being served as at the start, and two people are standing in the queue.) State the distribution of Elias' waiting time before his service commences. Give a brief explanation for your answer. Hence calculate the mean and standard deviation of Elias' waiting time. (d) Suggest two random processes associated with the queueing system, one with a discrete time domain and one with a continuous time domain. In your examples, you should use the notation for random processes introduced in Part I of Book 2; for example, {X(t);t≥0} or {X

n
​
;n=0,1,…}. Describe precisely what the random variables (X(t),X
n
​
or whatever) represent, and state clearly the state space corresponding to each process. (See Activity 2.5 in Book 2 for the sort of thing that is required here.)

Answer 1

The distribution of the time that Anita will have to wait before she can move forward for service is a gamma distribution. Her expected waiting time is 50 seconds. The probability that Anita will have to wait more than three minutes until an assistant is free to serve her can be calculated using the cumulative distribution function (CDF) of the gamma distribution.

Step-by-step explanation:

To find the distribution of the time that Anita will have to wait before she can move forward for service, we need to consider the service times of the four assistants. Since the service times are exponentially distributed, we know that the waiting times will also follow an exponential distribution. The sum of independent exponential random variables with different means follows a gamma distribution. In this case, since we have four assistants with means 2 minutes, 3 minutes, 5 minutes, and 6 minutes, the waiting time distribution can be represented as a gamma distribution with shape parameter k = 4 and rate parameter λ = 1/2 + 1/3 + 1/5 + 1/6.

To calculate the expected waiting time, we can find the mean of the gamma distribution, which is given by k/λ. Plugging in the values of k and λ, we get the expected waiting time to be 50 seconds.

To calculate the probability that Anita will have to wait more than three minutes until an assistant is free to serve her, we need to find the cumulative distribution function (CDF) of the gamma distribution and evaluate it at three minutes. This probability can be calculated using software or tables for the gamma distribution.

For part (c), since Anita has been waiting for two minutes, we know that the waiting time distribution for Elias will be the same as Anita's initial waiting time distribution, but shifted by two minutes. So if we represent Anita's initial waiting time distribution as a gamma distribution with parameters k' and λ', Elias' waiting time distribution will also be a gamma distribution with parameters k' and λ' but shifted by two minutes. The mean and standard deviation of Elias' waiting time can be calculated using the properties of the gamma distribution.

For part (d), a random process with a discrete time domain could be the arrival of customers at the bank. We can represent this process as {N(t); t≥0}, where N(t) is the number of customers that have arrived by time t. The state space for this process would be the non-negative integers, as the number of customers cannot be negative.

A random process with a continuous time domain could be the waiting time of customers in the queue. We can represent this process as {X(t); t≥0}, where X(t) is the waiting time of the customer at time t. The state space for this process would be the non-negative real numbers, as the waiting time can be any positive value.