1

Safety Capacity

Effect of Buffer Capacity

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

0.749 0.768 0.782

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Safety Capacity

Economics of Capacity Decisions

Cost of Lost Business Cb

– $ / customer

– Increases with competition

Cost of Buffer Capacity Ck

– $/unit/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

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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/minuteEffect 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

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Safety Capacity

Optimal Processing Capacity

c K = 6 – c Pb Ii TC ($/hr) = $20c + $5(K+c) + $100Ri 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

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Safety Capacity

Performance Variability

Effect of Variability– Average versus Actual Flow time

Time Guarantee – Promise

Service Level– P(Actual Time Time Guarantee)

Safety Time– Time Guarantee – Average Time

Probability Distribution of Actual Flow Time– P(Actual Time t) = 1 – EXP(- t / T)

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Safety Capacity

Effect of Blocking and Abandonment

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

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Safety Capacity

Net Rate: Ri(1- Pb)(1- Pa)

Throughput Rate:R=min[Ri(1- Pb)(1- Pa),Rp]

Effect of Blocking and Abandonment

8

Safety Capacity

Example 8.8 - DesiCom Call Center

Arrival Rate Ri= 20 per hour=0.33 per min

Processing time Tp =2.5 minutes (24/hr)Number of servers c=1Buffer 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 minAverage total time in the system T=7.44 minAverage total number of customers in the system I=2.29

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Safety Capacity

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

Example 8.8 - DesiCom Call Center

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Safety Capacity

Example 8.8 - DesiCom Call Center

Number of lines 5 6 7 8 9 10

Number of servers c 1 1 1 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

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Safety Capacity

Capacity Investment Decisions

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

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Safety Capacity

Example 8.9 - Effect of Buffer Capacity on Total Cost

Number of lines n 5 6 7 8 9

Number of CSR’s c 1 1 1 1 1

Buffer capacity K=n-c 4 5 6 7 8

Cost of servers ($/hr)=20c 20 20 20 20 20

Cost of tel.lines ($/hr)=5n 25 30 35 40 45

Blocking Probability Pb (%) 10.04 7.71 6.03 4.78 3.83

Lost margin = $100RiPb200.8 154.2 120.6 95.6 76.6

Average number of calls in queue Ii1.23 1.52 1.79 2.04 2.27

Hourly cost of waiting=120Ii147.6 182.4 214.8 244.8 272.4

Total cost of service, blocking and waiting ($/hr)

393.4 386.6 390.4 400.4 414

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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 RiPb

(number/hr)

Queue length

Ii

Total Cost ($/hour)

1 5 7.71% 1.542 1.52 30+20+(1.542x100)+(1.52x120)=386.6

2 4 0.43% 0.086 0.16 30+40+(0.086x100)+(0.16x120)=97.8

3 3 0.09% 0.018 0.02 30+60+(0.018x100)+(0.02x120)=94.2

4 2 0.04% 0.008 0.00 30+80+(0.008x100)+(0.00x120)110.8

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Safety Capacity

Variability in Process Performance

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.

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Safety Capacity

Example 8.11 - WalCo Drugs

One pharmacist, DaveAverage of 20 customers per hourDave takes Average of 2.5 min to fill prescriptionProcess rate 24 per hourAssume 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

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Safety Capacity

Example 8.11 - Probability distribution of the actual time customer spends in process

(obtained by simulation)

0

2000

4000

6000

8000

10000

12000

14000

Total Time in Process

Fre

qu

ency

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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,15min

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Safety Capacity

Service Promise:Tduedate , Service Level & Safety Time

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.

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Safety Capacity

Due Date Quotation

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)

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Safety Capacity

Example 8.12 - WalCo Drug

WalCo has set SL = 0.95Assuming total time for customers is exponential

Tduedate = -T ln(1 – SL)

Tduedate = -T ln(0.05) = 3TFlow time for 95 percentile of exponential distribution is three times

the average T

Tduedate = 3 * 15 = 4595% of customers will get served within 45 min

Tduedate = T + Tsafety

Tsafety = 45 – 15 = 30 min30 min is the extra margin that WalCo should allow as protection

against variability

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Safety Capacity

Relating Utilization and Safety Time: Safety Time Vs. Capacity Utilization

Capacity utilization ρ 60 % 70% 80% 90%

Waiting time Ti 1.5Tp 2.33Tp 4Tp 9Tp

Total flow time T= Ti + Tp 2.5Tp 3.33Tp 5Tp 10Tp

Promised time Tduedate 7.7Tp 10Tp 15Tp 30Tp

Safety time Tsafety = Tduedate – T 5Tp 6.67Tp 10Tp 20Tp

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

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Safety Capacity

Managing Customer Perceptions and Expectations

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

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Safety Capacity

Thank you

Questions?