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Problem 8.4 K = ∞ R i = 60/4 = 15 /hr T p = 3 min = 0.05 hour c= 1 R p = c/Tp = 20 /hour

Calculate total hourly costs, customer service proportion, and waiting time with varying server and line configurations to maximize efficiency and minimize expenses. Analyze the economic impact of adding servers and lines.

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Problem 8.4 K = ∞ R i = 60/4 = 15 /hr T p = 3 min = 0.05 hour c= 1 R p = c/Tp = 20 /hour

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  1. Problem 8.4 K = ∞ Ri = 60/4 = 15 /hr Tp = 3 min = 0.05 hour c= 1 Rp = c/Tp = 20 /hour Both Ti and Tp exponential Server $20 /hr Phone $5/hr Wait cost $120/person/hr a4) the total hourly cost of service and waiting 75% of time CSR will be busy a2) the proportion of time that CSR will be busy a3) the average number of customers on line

  2. Problem 8.4 K = ∞ Ri = 60/4 = 15 /hr Tp = 3 min = 0.05 hour c= 1 Rp = c/Tp = 20 /hour Both Ti and Tp exponential Server $20 /hr Phone $5/hr Wait cost $120/person/hr We know the formula for Ii, what about Ip? Why ρ customers in the server? In this example with ρ = 0.75 Suppose we check the CSR at 100 random times. On average on how many times is s/he is talking to a customer? 75% one customer, 25% no customer. On average 0.75 customer are in the server. In a single server ρ customer are served, in multi server cρ customers Ip = c ρ a4) the total hourly cost of service and waiting The costs include the CSR wages and the cost of waiting (line charge + waiting cost for customers). Service cost = 20+5 = 25/hr Waiting cost = 2.25(120) = 270/hr Total cost = 295 /hr

  3. b) Compute service cost, waiting cost, blocking cost K = ∞  K= 4 With only four lines and one CSR,  c = 1, Maximum buffer size K = 3. Using the spreadsheet Performance.xls Average waiting time Ti = 3.45 mins, Average # of customers on hold Ii = 0.77, (in K = ∞, Ii was 2.25) Average number of customers in system is still I = 1.44, Probability of blocking = 0.104. Ip is still equal to ρ but ρ is not .75 any more In this case the costs incurred are the CSR wages, the cost of waiting (line charge + waiting cost for customers) and the lost business because of blocked calls. We have Hourly wages of CSR = $20 / hour, Line charge = $5 / hour, Customer waiting cost = .77120 = $92.4, Cost of blocking = Calls blocked per hour$100 = Probability of blockingAverage arrival rate$100 = 0.10415$100 = $156. This implies that Total hourly cost = $20 + $5 + $92.4 + $156 = $273.4.

  4. c1) How would it affect customer waiting time? C2) What is the economic impact of adding another line. Adding one line  K = 4 all other parameters remain the same. Using the Performance.xls spreadsheet Ti = 4.33 mins, Ii = 1.005, Average number of customers in system I = 1.70, Probability of blocking = 0.072. Customer waiting cost = 1.005120 = $120.6, Cost of blocking = Calls blocked per hour$100 = Probability of blockingAverage arrival rate$100 = 0.07215$100 = $108. Excluding the cost of the new line we have Total cost per hour = $20 + $5 + $120.6 + $108 = $253.6. As long as the cost of the new line is less than $273.4 (cost with 4 lines) - $253.6 (cost with 3 lines) = $19.8 / hour, it pays to install the new line.

  5. d) Beside adding a new line, also add a CSR. Two servers and five lines Average arrival rate Ri = 1/4 per minute, Average unit capacity 1/Tp = 1/3 per minute, Number of servers c = 2, Maximum buffer size K = 3. Using the Performance.xls spreadsheet we get Average waiting time Ti = 0.42 mins, Average # of customers on hold Ii = 0.105 Average number of customers in system I = 0.85, Probability of blocking = 0.007. Service cost = 40 + 5 / hour, Waiting cost =.105120 = $12.6, Blocking cost = Probability of blocking  Average arrival rate$100 = 0.00715$100 = $10.5. Hourly cost of system = $40 + $5 + $12.6 + $10.5 = $68.1/hour. This is a significant reduction in cost. The new CSR should thus be hired.

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