Question

1. An average of 10 cars per hour arrive at a single-server drive-in teller. Assume that the average service time for each customer is 4 minutes, and both interarrival times and service times are exponential.

   1. What is the arrival rate per minute?

   2. What is the servicing rate per minute?

   3. What is the servicing rate per hour?

   4. What is the traffic intensity?

   5. What is the probability that the teller is idle?

   6. What is the average number of cars waiting in line for the teller?

   7. What is the average number of cars in the drive-in facility (waiting or serviced)?

   8. What is the average amount of time a drive-in customer spends in the facility (waiting

      or being serviced?

   9. What is the average amount of time a customer spends in the waiting line?

Answer 1

Solution

This is a direct application of M/M/1 Queue system.

Back-up Theory

An M/M/1 queue system is characterized by arrivals following Poisson pattern with average rate λ, [this is also the same as exponential arrival with average inter-arrival time = 1/λ] service time following Exponential Distribution with average service time of (1/µ) [this is also the same as Poisson service with average service rate = µ] and single service channel.

Let n = number of customers in the system and m = number of customers in the queue.

[Trivially, n = m + number of customers under service.]

Traffic Intensity = ρ = (λ/µ)…………………………………………………………...........………..(A)

The steady-state probability of n customers in the system is given by P_n = ρⁿ(1 - ρ) …………(1)

The steady-state probability of no customers in the system is given by P₀ = (1 - ρ) …….……(2)

Average queue length = E(m) = (λ²)/{µ(µ - λ)} ……………………………………….......…..…..(3)

Average number of customers in the system = E(n) = (λ)/(µ - λ)………………...……………..(4)

Average waiting time = E(w) = (λ)/{µ(µ - λ)} …………………………………….........…………..(5)

Average time spent in the system = E(v) = {1/(µ - λ)}………………………….....……………..(6)

Now to work out the solution,

Given λ = 10 per hour and µ = 15 per hour [service time is 4 minutes => 15 customers per 60 minutes] ..................... (7)

1) Arrival rate per minute = 10/60 = 0.1667 Answer 1 [vide (7)]

2) Servicing rate per minute = 15/60 = 0.25 Answer 2 [vide (7)]

3) Servicing rate per hour 15 Answer 3 [vide (7)]

4) Traffic intensity = ρ = 10/15 = 0.6667 Answer 4 [vide (A) and (7)]

5) Probability that the teller is idle = P₀ = (1 - ρ) = 0.3333 Answer 5 [vide (2) and (A)]

6) Average number of cars waiting in line for the teller     = E(m) = 100/(15 x 5) = 1.3333 Answer 6 [vide (3) and (7)]

7) Average number of cars in the drive-in facility = E(n) = 15/5 = 3 Answer 7 [vide (4) and (7)]

8) Average amount of time a customer spends in the facility = E(v) = 1/5 hour = 12 minutes Answer 8 [vide (6) and (7)]

9) Average amount of time a customer spends in the waiting line     = E(w) = 10/(15 x 5) = 2/15 hour = 8 minutes Answer 9 [vide (5) and (7)]

DONE