Question

In a queueing system there is one server. Customers arrive at the queueing system according to a Poisson process with arrival rate 20 customers per hour. An arriving customer finding n people in the system, immediately leaves with probability

qₙ=n/4, n=0,1....4
Customers that join the system are served in order of arrival and the service times are assumed to be independent and exponentially distributed with a mean service time equal to 3 minutes.
Let X (t) denote the number of customers in the system at time t. Assume that X(0) = 0.
Explain why X(t) is a birth and death process and give the birth and death rates.
What is the probability that the first customer has departed the queue- ing system before the next customer arrives?

Answer 1

X(t) is a birth and death process with a birth rate of 20 customers per hour and a death rate of 1/3 customers per minute. The probability that the first customer has departed before the next one arrives is approximately 0.3935.

Step-by-step explanation:

The number of customers in the queueing system at any given time, X(t), can be modeled as a birth and death process because it can increase (birth) when a customer arrives and decrease (death) when a customer departs. In this case, the birth rate is equal to the arrival rate of customers, which is 20 customers per hour.

The death rate can be calculated by multiplying the probability that a customer leaves immediately upon arrival, q_n, by the service rate, which is the reciprocal of the mean service time, 1/3 customers per minute.

To calculate the probability that the first customer has departed before the next one arrives, we can use the memorylessness property of the exponential distribution.

The memorylessness property states that the conditional probability of an event happening in the next time period is the same regardless of how much time has already elapsed. In this case, the time until the next customer arrives is exponentially distributed with a mean of 2 minutes (since the arrival rate is 30 customers per hour, or one customer every 2 minutes on average).

Therefore, the probability that the next customer arrives within 1 minute is 1 - e^(-1/2) ≈ 0.3935.