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 8 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 u will continue to be 7.5 calls per hour for next year's queueing system, determine L, La, W, and Wa 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 u will be adjustable when the number of technical representatives is reduced to one. Solve algebraically for the value of u that would vield the same value of W Repeat part (b) with Wa instead of W. as for the current svstem c) I need full answers with formulas please.

Answer 1

Answer:

Company PS provides technical assistance to the customers on a new software package Currently, it employs two customer representatives with the mean service time of 8 minutes. Calls arrive at a mean rate of 10 per hour The arrival rate of the customers would reduce to 5 per hour in the next year. Therefore, the company is planning to reduce the number of representatives to 1. One needs to determine the L, Ly Wand W, for both the systems.

The following information are provided: Case 1: service rate ( μ) : 7.5 customer per hour (60 minutes per hour 8 min per customer) Arrival rate ( λ) = 10 per hour Number of servers (s)- 2 Initially determine the value of utilization ( ρ) using given formula 10 2x7.5 = 0.67

Determine the value of LLW and using the given formulas in Excel spreadsheet: 1 Inputs Arrival rate (lambda) 3 customers/hour customershour servers Service rate (mu) 7.5 no. of identical servers (s) 6 Outputs Probability of no customers in system (PO) POinf(84,82/83 8 Avg, utilization rate of server 9 Avg. no. of customers waiting in line (Lq) 10 Avg. no. of customers in system (L) 11 Avg. time waiting in line (Wa) 12 Avg. time in system (W) 82/B3/B4 (B2/B3)(B4+1yB4/FACT(B4)(1-B2/B3/B4*2 'B7 customers -B9+B2/B3 customers hour hour -810/B2 1 2

The obtained values are shown below 1 Inputs Arrival rate (lambda) 3 Service rate (mu) 4 no. of identical servers (s) 10 customers/hour 7.5 customers/hour serverS 6 Outputs Probability of no customers in system (PO) Ava utilization rate of server Avg. no. of customers waiting in line (Lq) Avg. no. of customers in system (L) Avg. time waiting in line (Wq) Avg. time in system 0.20000 66.7% 1.06667 customers 2.40000 customers 0.10667 hour 0.24000 hour

Case 2: Service rate (-7.5 customer per hour (60 minutes per hour/ 8 min per customer) Mean arrival rate for next year ( λ)-5 per hour Number of servers (s)1 Determine the value of utilization ( ρ ) using given formula: x 7.5 -0.67

Determine the value of L.LW andWfor next year by changing the value of arrival rate (Cell B2) and number of identical servers (Cell B4) in the above Excel spreadsheet. btained values are shown below: 1 Inputs Arrival rate (lambda) 3 Service rate (mu) 4 no. of identical servers (s) customers/hour 7.5 customers/hour serverS 6 Outputs Probability of no customers in system (PO) Ava utilization rate of server Avg. no. of customers waiting in line (Lq) Avg. no. of customers in system (L) Avg. time waiting in line (Wq) Avg. time in system 0.33333 66.7% 1.33333 customers 2.00000 customers 0.26667 hour 0.40000 hour

Comparing the outputs of both the cases it is seen that the next year's system yields smaller values of L; however, the values of L,Wand w are higher for next year as compared with that of the current year values.

W for the current year 0.24 hour Mean arrival rate for next year ( λ)-5 per hour Number of servers (s)1 One needs to determine the value of Service rate ( μ) that would provide W-0.24 hours. Use the given formula: (μ-λ) +5 0.24 4.17+5 -9.17 customers per hour Hence, it is concluded that the required service rate ie. μ is 9.17 customers per hour.

Consider that the value of μ is adjustable. Now, find out the value of μ that would yield the same value of W" as that of the current year Consider the following values: W for the current year-0.10667 hour Mean arrival rate for next year ( λ)-5 per hour Number of servers (s)1 One needs to determine the value of service rate ( μ ) that would provide W-0.10667 hour. Use the given formula

The above equation is in the format of standard quadratic equation ax2 +bx+c0 The value of μ can be obtained using the given quadratic formula. 20 Substitute the quadratic equation formula in above equation (1) as shown below: Here x=μ a= W

Therefore μ= 2W 5x0.107v(5x0.107)+4x0.107x:5 0.5350.286+2.14 0.535+1.558 2x0.107 0.214 0.214 0.5351.558 0.535-1.558 0.214 0.214 9.78,-4.78 Since, the service rate cannot be negative. Therefore, the required service rate ie. customers per hour μ is 9.78