# This problem calculates average queue size and waiting time.

ABC Airline has one reservation clerk on duty at a time. He handles information about flight schedules and makes reservations. All calls to ABC Airline are answered by an operator. If a caller requests information or reservations, the operator transfers the call to the reservations clerk. If the clerk is busy, the operator asks the caller to wait. When the clerk becomes available, the operator transfers to him the call of the person who has been waiting the longest. Assume that arrivals and services follow a Poisson process. Calls arrive at a rate of 10 per hour (Î» = 10 per hour), and the reservations clerk can service a call in four minutes on average (Î¼ = 15 per hour).

Part 1: What is the average number of calls waiting to be connected to the reservations clerk?

Part 2: What is the average time a caller must wait before reaching the reservations clerk?

Part 3: What is the average time for a caller to complete a call (waiting time plus service time)?

https://brainmass.com/business/operations-research/problem-calculates-average-queue-size-waiting-time-614508

## SOLUTION This solution is **FREE** courtesy of BrainMass!

Solution

This is a single-server queuing scenario. Common notation is:

Î» = Average number of customers arriving in one unit of time

Î¼ = Average number of customers the facility is capable of servicing in one unit of time, assuming no shortage of customers

Lq = Expected number in the queue (the number in the queue does not include the unit being serviced)

Wq = Expected time an arrival must wait in the queue

Part 1:

L_q=Î»^2/(Î¼(Î¼-Î»))=ã€–10ã€—^2/(15(15-10))=100/75=1.33

Part 2:

W_q=L_q/Î»=1.33/10=0.133 hours or 8 minutes

Part 3:

Expected total time = Wq + expected service time = 8 + 4 = 12 minutes

https://brainmass.com/business/operations-research/problem-calculates-average-queue-size-waiting-time-614508