# Find a Doctor (FaD) is a small startup that helps people find a physician that best meets their...

## Question:

Find a Doctor (FaD) is a small startup that helps people find a physician that best meets their needs (location, insurance accepted, etc.) During a "slow" time for FaD 7, staff members take calls from customers. On average, one call arrives every 5 minutes. On average, each staff member spends 20 minutes with each customer. How Long (in minutes) does a customer spend on average waiting on hold before he or she can start speaking to a representative?

a. 1.2 mins

b. 0.7 mins

c. 11.4 mins

d. 15.2 mins

## Queue

Queue refers to the line of customer waiting for the consumption of service of a partcular product. It is the line of people thoes are waiting for their chance of seeking the service from the service render or producer.

The option a. 1.2 mins closest, therefore, it is the correct answer.

{eq}\begin{align*} {\rm{Utilization}} &= \dfrac{{{\rm{Arrival}}\,{\rm{Rate}}}}{{{\rm{Service}}\,{\rm{Rate \times Number}}\,{\rm{Of}}\,{\rm{Servers}}}}\\ &= \dfrac{{1/5}}{{1/20 \times 7}}\\ &= \dfrac{{0.2}}{{0.05 \times 7}}\\ &= \dfrac{{0.2}}{{0.35}}\\ &= 0.6\% \,(approx) \end{align*} {/eq}

The average waiting time for the customers assuming that the coefficients of variation for the arrival process and the service process are both 1.

{eq}\begin{align*} {\rm{Time}} &= \left[ {\dfrac{{{\rm{Processing}}\,{\rm{Time}}}}{{{\rm{Number}}\,{\rm{Of}}\,{\rm{Server}}}}} \right] \times \left[ {\dfrac{{{\rm{Utilizatio}}{{\rm{n}}^{\sqrt {2[(m + 1)} - 1}}}}{{{\rm{1 - }}\,{\rm{Utilization}}}}} \right] \times \left[ {\dfrac{{{\rm{CO}}{{\rm{V}}_{\rm{a}}}^2{\rm{ + CO}}{{\rm{V}}_{\rm{p}}}^2}}{{\rm{2}}}} \right]\\ &= \left[ {\dfrac{{20}}{7}} \right] \times \left[ {\dfrac{{{{0.6}^3}}}{{1 - 0.6}}} \right] \times \left[ {\dfrac{{1 + 1}}{2}} \right]\\ &= 2.8 \times \dfrac{{0.216}}{{0.4}} \times 1\\ &= 2.8 \times 0.54 \times 1\\ &= 1.512 \end{align*} {/eq} 