# What is the fraction of call attempts that receive a busy signal?

###### Cost Management.
December 24, 2020
###### Marketing Management.
December 24, 2020

A small store has a single phone line. Calls to the store arewell modeled by an M / M /1 retrial queue with arrival rate ʎ = 10per hour, service rate µ = 15 per hour, and retrial rate  β= 6 per hour. (a) What is the average length of time for a customer to reach aclerk at the store? (b) What is the average rate that call attempts are made to thestore? (Call attempts include calls that are answered and calls thatreceive a busy signal.) (c) What is the fraction of call attemptsthat receive a busy signal? (d) Suppose that you are measuring callattempts to the store and you are not aware that some customers aremaking redial attempts. That is, you assume each call attempt isfrom a distinct customer. You decide to model the system as anM/M/l/l queue, where the arrival rate is the rate of call attemptsfound in (b). Based on these assumptions, what is the fraction ofcalls that receive a busy signal? Compare this to the actual resultfound in (c)