Question

People's Software Company has just set up a call center to provide technical assistance on its new software package. Two technical representatives are taking the calls, where the time required by either representative to answer a customer's questions has an exponential distribution with a mean of 5 minutes. Calls are arriving according to a Poisson process at a mean rate of 10 per hour. By next year, the mean arrival rate of calls is expected to decline to 5 per hour, so the plan is to reduce the number of technical representatives to one then. a-) Assuming that service rate μ will stay the same for next year's queueing system, determine L, Lq, W, and Wq for both the current system and next year's system. For each of these four measures of performance, which system yields the smaller value? b-) Now assume that μ will be adjustable when the number of technical representatives is reduced to one. Solve algebraically for the value of μ that would yield the same value of W as for the current system.

Answer 1

9.17

9.78

Explanation:

Solution:-

a)

The following information is provided:

Case 1:

- Service rate ( μ ) = 12 customer per hour ( 60 minutes per hour / 5 minutes time per customer )

- Arrival rate ( λ ) = 10 per hour

- Number of servers ( s ) = 2

- Initially determine the value of utilization ( ρ ):

ρ = λ / s*μ

ρ = 10 / 2*12

ρ = 0.4167 - 41.67%

- Determine the values of L, Lq , W and Wq. using the given formulas in the spreadsheet attached. Figure 1

- Figure 2 expresses the result of all the calculations.

Case 2:

- Service rate ( μ ) = 12 customer per hour ( 60 minutes per hour / 5 minutes time per customer )

- Arrival rate ( λ ) = 5 per hour

- Number of servers ( s ) = 1

- Initially determine the value of utilization ( ρ ):

ρ = λ / s*μ

ρ = 5 / 1*12

ρ = 0.4167 - 41.67%

- Determine the values of L, Lq , W and Wq. using the given formulas in the spreadsheet attached. Figure 3, expresses the results

- Comparing the outputs of both the cases we see that next year's system yields smaller values of L; however, the values of Lq, W ,Wq are higher for next year as compared to last years.

W for the current year = 0.24

Mean arrival rate ( λ ) = 5 per hour

Number of servers ( s ) = 1

- Determine the value of service rates ( μ ) that would provide W = 0.24 hrs. Use the given formula:

W = 1 / ( μ - λ )

μ = 1 / W + λ

μ = 1 /0.24 + 5

μ = 9.17 customers per hour

- Hence, its concluded that the required service rate is 9.17 customers per hour.

c)

Consider the value of μ is adjustable now. Now find the value of μ such that it takes the value of "Wq" from current year.

Wq for the current year = 0.10667 hours

Mean arrival rate ( λ ) = 5 per hour

Number of servers ( s ) = 1

- Determine the value of service rates ( μ ) that would provide Wq = 0.10667 hrs. Use the given formula:

Wq = λ / μ*( μ - λ )

Wq*μ^2 - Wq*λ*μ - λ = 0

0.10667*μ^2 - 0.53335*μ - 5 = 0

- Solve the quadratic and obtain values of (μ):

μ = 9.78 , -4.78

- The negative value of μ is discarded while the positive value μ = 9.78 customers per hours is the required service rate.