Safety Capacity Effect of Buffer Capacity 1 Process

Safety Capacity Effect of Buffer Capacity 1 Process Data – Ri = 20/hour, Tp = 2. 5 mins, c = 1, K = # Lines – c Performance Measures K 4 5 6 Ii 1. 23 1. 52 1. 79 Ti 4. 10 4. 94 5. 72 Pb 0. 1004 0. 0771 0. 0603 R 17. 99 18. 46 18. 79 r 0. 749 0. 768 0. 782

Safety Capacity Economics of Capacity Decisions 2 Cost of Lost Business Cb – $ / customer – Increases with competition Cost of Buffer Capacity Ck – $/unit time Cost of Waiting Cw – $ /customer/unit time – Increases with competition Cost of Processing Cs – $ /server/unit time – Increases with 1/ Tp Tradeoff: Choose c, Tp, K – Minimize Total Cost/unit time = Cb Ri Pb + Ck K + Cw I (or Ii) + c Cs

Safety Capacity Optimal Buffer Capacity Cost Data – Cost of telephone line = $5/hour, Cost of server = $20/hour, Margin lost = $100/call, Waiting cost = $2/customer/minute Effect of Buffer Capacity on Total Cost K $5(K + c) $20 c $100 Ri Pb $120 Ii TC ($/hr) 4 25 20 200. 8 147. 6 393. 4 5 30 20 154. 2 182. 6 386. 4 6 35 20 120. 6 214. 8 390. 4 3

Safety Capacity Optimal Processing Capacity 4 c K=6–c Pb Ii TC ($/hr) = $20 c + $5(K+c) + $100 Ri Pb+ $120 Ii 1 5 0. 0771 1. 542 $386. 6 2 4 0. 0043 0. 158 $97. 8 3 3 0. 0009 0. 021 $94. 2 4 2 0. 0004 0. 003 $110. 8

Safety Capacity Performance Variability 5 Effect of Variability – Average versus Actual Flow time Time Guarantee – Promise Service Level – P(Actual Time Guarantee) Safety Time – Time Guarantee – Average Time Probability Distribution of Actual Flow Time – P(Actual Time t) = 1 – EXP(- t / T)

Safety Capacity Effect of Blocking and Abandonment 6 Blocking: the buffer is full = new arrivals are turned away Abandonment: the customers may leave the process before being served Proportion blocked Pb Proportion abandoning Pa

Safety Capacity Effect of Blocking and Abandonment 7 Net Rate: Ri(1 - Pb)(1 - Pa) Throughput Rate: R=min[Ri(1 - Pb)(1 - Pa), Rp]

Safety Capacity Example 8. 8 - Desi. Com Call Center 8 Arrival Rate Ri= 20 per hour=0. 33 per min Processing time Tp =2. 5 minutes (24/hr) Number of servers c=1 Buffer capacity K=5 Probability of blocking Pb=0. 0771 Average number of calls on hold Ii=1. 52 Average waiting time in queue Ti=4. 94 min Average total time in the system T=7. 44 min Average total number of customers in the system I=2. 29

Safety Capacity Example 8. 8 - Desi. Com Call Center 9 Throughput Rate R=min[Ri(1 - Pb), Rp]= min[20*(1 -0. 0771), 24] R=18. 46 calls/hour Server utilization: R/ Rp=18. 46/24=0. 769

Safety Capacity Example 8. 8 - Desi. Com Call Center 10 Number of lines 5 6 7 8 9 10 Number of servers c 1 1 1 Buffer Capacity K 4 5 6 7 8 9 Average number of calls in queue 1. 23 1. 52 1. 79 2. 04 2. 27 2. 47 Average wait in queue Ti (min) 4. 10 4. 94 5. 72 6. 43 7. 08 7. 67 Blocking Probability Pb (%) 10. 04 7. 71 6. 03 4. 78 3. 83 3. 09 Throughput R (units/hour) 17. 99 18. 46 18. 79 19. 04 19. 23 19. 38 Resource utilization . 749 . 769 . 782 . 793 . 801 . 807

Safety Capacity Investment Decisions 11 The Economics of Buffer Capacity Cost of servers wages =$20/hour Cost of leasing a telephone line=$5 per line per hour Cost of lost contribution margin =$100 per blocked call Cost of waiting by callers on hold =$2 per minute per customer Total Operating Cost is $386. 6/hour

Safety Capacity Example 8. 9 - Effect of Buffer Capacity on Total Cost 12 Number of lines n 5 6 7 8 9 Number of CSR’s c 1 1 1 Buffer capacity K=n-c 4 5 6 7 8 Cost of servers ($/hr)=20 c 20 20 20 Cost of tel. lines ($/hr)=5 n 25 30 35 40 45 Blocking Probability Pb (%) 10. 04 7. 71 6. 03 4. 78 3. 83 Lost margin = $100 Ri. Pb 200. 8 154. 2 120. 6 95. 6 76. 6 Average number of calls in queue Ii 1. 23 1. 52 1. 79 2. 04 2. 27 Hourly cost of waiting=120 Ii 147. 6 182. 4 214. 8 244. 8 272. 4 Total cost of service, blocking and waiting ($/hr) 393. 4 386. 6 390. 4 400. 4 414

Safety Capacity Example 8. 10 - The Economics of Processing Capacity The number of line is fixed: n=6 The buffer capacity K=6 -c c K Blocking Pb(%) Lost Calls Ri. Pb (number/hr) Queue length Ii Total Cost ($/hour) 1 5 7. 71% 1. 542 1. 52 30+20+(1. 542 x 100)+(1. 52 x 120)=386. 6 2 4 0. 43% 0. 086 0. 16 30+40+(0. 086 x 100)+(0. 16 x 120)=97. 8 3 3 0. 09% 0. 018 0. 02 30+60+(0. 018 x 100)+(0. 02 x 120)=94. 2 4 2 0. 04% 0. 008 0. 00 30+80+(0. 008 x 100)+(0. 00 x 120)110. 8 13

Safety Capacity Variability in Process Performance 14 Why considering the average queue length and waiting time as performance measures may not be sufficient? Average waiting time includes both customers with very long wait and customers with short or no wait. We would like to look at the entire probability distribution of the waiting time across all customers. Thus we need to focus on the upper tail of the probability distribution of the waiting time, not just its average value.

Safety Capacity Example 8. 11 - Wal. Co Drugs 15 One pharmacist, Dave Average of 20 customers per hour Dave takes Average of 2. 5 min to fill prescription Process rate 24 per hour Assume exponentially distributed interarrival and processing time; we have single phase, single server exponential model Average total process is; T = 1/(Rp – Ri) = 1/(24 -20) = 0. 25 or 15 min

Safety Capacity Example 8. 11 - Probability distribution of the actual time customer spends in process (obtained by simulation) 16

Safety Capacity Example 8. 11 - Probability Distribution Analysis 65% of customers will spend 15 min or less in process 95% of customers are served within 40 min 5% of customers are the ones who will bitterly complain. Imagine if they new that the average customer spends 15 min in the system. 35% may experience delays longer than Average T, 15 min 17

Safety Capacity Tduedate Service Promise: , Service Level & Safety Time 18 SL; The probability of fulfilling the stated promise. The Firm will set the SL and calculate the Tduedate from the probability distribution of the total time in process (T). Safety time is the time margin that we should allow over and above the expected time to deliver service in order to ensure that we will be able to meet the required date with high probability Tduedate = T + Tsafety Prob(Total time in process <= Tduedate) = SL Larger SL results in grater probability of fulfilling the promise.

Safety Capacity Due Date Quotation 19 Due Date Quotation is the practice of promising a time frame within which the product will be delivered. We know that in single-phase single server service process; the Actual total time a customer spends in the process is exponentially distributed with mean T. SL = Prob(Total time in process <= Tduedate) = 1 – EXP( - Tduedate /T) Which is the fraction of customers who will no longer be delayed more than promised. Tduedate = -T ln(1 – SL)

Safety Capacity Example 8. 12 - Wal. Co Drug 20 Wal. Co has set SL = 0. 95 Assuming total time for customers is exponential Tduedate = -T ln(1 – SL) Tduedate = -T ln(0. 05) = 3 T Flow time for 95 percentile of exponential distribution is three times the average T Tduedate = 3 * 15 = 45 95% of customers will get served within 45 min Tduedate = T + Tsafety = 45 – 15 = 30 min is the extra margin that Wal. Co should allow as protection against variability

Safety Capacity Relating Utilization and Safety Time: Safety Time Vs. Capacity Utilization 21 Capacity utilization ρ Waiting time Ti Total flow time T= Ti + Tp Promised time Tduedate Safety time Tsafety = Tduedate – T 60 % 1. 5 Tp 2. 5 Tp 7. 7 Tp 5 Tp 70% 2. 33 Tp 3. 33 Tp 10 Tp 6. 67 Tp 80% 4 Tp 5 Tp 10 Tp 90% 9 Tp 10 Tp 30 Tp 20 Tp Higher the utilization; Longer the promised time and Safety time Safety Capacity decreases when capacity utilization increases Larger safety capacity, the smaller safety time and therefore we can promise a shorter wait

Safety Capacity Managing Customer Perceptions and Expectations 22 Uncertainty about the length of wait (Blind waits) makes customers more impatient. Solution is Behavioral Strategies Making the waiting customers comfortable Creating distractions Offering entertainment

Safety Capacity Thank you 23 Questions?