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 relatively “slow” time for them, they have 5 staff members in a call center taking calls from customers. On average, one call arrives every 6 minutes with a standard deviation of 6 minutes. Each staff member spends 20 minutes with each customer on average, with a standard deviation of 30 minutes

A.  What is the probability that one of their staff members is busy (as a %), i.e., what percentage of time will the staff members be busy?

B. How long (in minutes) does a customer spend on average waiting on hold before they can start speaking to a representative from FaD?

C. What is the average number of callers who are in FaD’s call center system (either waiting or being served by the operator)? [Note: Average number of customers in the system does not need to be an integer because it is an average over time.]

D. Suppose that for every minute of time a caller waits on hold for service the call center is required to pay out a penalty of 75 cents. Estimate the hourly penalty pay out by the call center for caller wait times. [FYI, the cost of hiring a new qualified call center operator (identical to the current operators) is $20 per hour. Estimating the hourly penalty pay out allows you to decide if increasing the number of operators will reduce total staffing + penalty pay out costs.]

E. If you were asked to list mechanisms to improve the call center, what two things would you target for improvement and how might the improvement be achieved?

F.  Calculate the value of cd 2 , the SCV (squared coefficient of variation) of the inter-departure times from the system.

G. ) Suppose the call center operator always works through a computer technology interface, which is prone to random, preemptive breakdown. [When the technology is down, the operator also cannot work.] The computer technology resource is subject to random failure with mean time to failure (MTTF or uptime), mf = 8 hours, and mean time to repair (MTTR or downtime), mr = 20 minutes. Repair time is exponentially distributed with SCV, cr 2 = 1. Calculate the mean and SCV of the effective call processing time, te and ce 2 , after incorporating computer technology breakdown.

Answer 1

A. Utilization of each staff = u = Avg. activity time / Avg. interarrival time = 20 / (5*6) = 0.667 or 66.67%

B.

CVa = Coefficient of variation of arrival = SD_arrival / Avg. arrival time = 6/6 = 1

CVp = Coefficient of variation of activity = SD_process / Avg. process time = 30/20 = 1.5

m = number of servers = 5

Activity time = 20 min

Utilization = 0.6667

Formula for waiting time is

Activity time. utilizatio n./2(m+D-1)(CVỈ+CIm 1 - utilizatio n 9

Time in queue = Tq = 7.18 min

C.

Avg. no. of calls waiting = Inventory waiting = Iq = Tq / Avg. arrival time = 7.18 / 6 = 1.2

Avg. no. of calls in process = Inventory in process = Ip = m*utilization = 5*0.6667 = 3.33

Total calls waiting = 1.2 + 3.33 = 4.53

D.

Avg. waiting time = 7.18 min for one caller

per hour Avg. no. of callers = 60/6 = 10

Total waiting minutes per hour = 71.8 minutes

Penalty = $ 71.8*0.75 per hours = $54.85