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 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

We know the formula for I i, what about I p ? 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. 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 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 Both Ti and Tp exponential Server $20 /hr Phone $5/hr Wait cost $120/person/hr In a single server ρ customer are served, in multi server cρ customers I p = c ρ

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 = 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 =  15  $100 = $156. This implies that Total hourly cost = $20 + $5 + $ $156 = $273.4.

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 = Customer waiting cost =  120 = $120.6, Cost of blocking = Calls blocked per hour  $100 = Probability of blocking  Average arrival rate  $100 =  15  $100 = $108. Excluding the cost of the new line we have Total cost per hour = $20 + $5 + $ $108 = $ 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.

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 = Average number of customers in system I = 0.85, Probability of blocking = Service cost = / hour, Waiting cost =.105  120 = $12.6, Blocking cost = Probability of blocking  Average arrival rate  $100 =  15  $100 = $10.5. Hourly cost of system = $40 + $5 + $ $10.5 = $68.1/hour. This is a significant reduction in cost. The new CSR should thus be hired.