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1Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Made-to-stock (MTS) operations Product is manufactured and stocked in advance of
demand Inventory permits economies of scale and protects
against stockouts due to variability of inflows and outflows
Make-to-order (MTO) process Each order is specific, cannot be stored in advance Process manger needs to maintain sufficient capacity Variability in both arrival and processing time Role of capacity rather than inventory Safety inventory vs. Safety Capacity Example: Service operations
Make to stock vs. Make to Order
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2Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Banks (tellers, ATMs, drive-ins) Fast food restaurants (counters, drive-ins) Retail (checkout counters) Airline (reservation, check-in, takeoff, landing,
baggage claim) Hospitals (ER, OR, HMO) Service facilities (repair, job shop, ships/trucks
load/unload) Some production systems- to some extend (Dell
computer) Call centers (telemarketing, help desks, 911
emergency)
Examples
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3Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
The DesiTalk Call Center
Sales RepsProcessing
Calls
(Service Process)
Incoming Calls(Customer Arrivals) Calls
on Hold(Service Inventory)
Answered Calls(Customer Departures)
Blocked Calls(Due to busy signal)
Abandoned Calls(Due to long waits)
Calls In Process(Due to long waits)
The Call Center Process
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4Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Tp : processing time. Tp units of time. Example, on average it takes 5
minutes to serve a customer. Rp : processing rate. Rp flow units are handled per unit of time. If Tp is 5 minutes. Compute Rp. Rp= 1/5 per minute. Or 60/5 = 12 per hour.
Service Process Attributes
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5Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Tp : processing time. Rp : processing rate. What is the relationship between Rp and Tp?If we have one resource Rp = 1/Tp
Service Process Attributes
What is the relationship between Rp and TpRp = c/Tp
Each customer always stay for Tp unites of time with the server
In general when we have c recourses
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6Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Tp = 5 minutes. Processing time is 5 minute. Each customer on average is with the server for 5 minutes.
c = 3, we have three servers.Processing rate of each server is 1/5 customers per
minute. Or 12 customer per hour.Rp is the processing rate of all three servers.Rp = c/Tp Rp = 3/5 customers/minute. Or 36 customers/hour.
Service Process Attributes
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7Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Service Process Attributes
Ta : customer inter-arrival time.
On average each 10 minutes one customer arrives. Ri : customer arrival (inflow) rate.
What is the relationship between Ta and RiTa = every ten minutes one customer arrivesHow many customers in a minute? 1/10; Ri = 1/Ta= 1/10Ri = 1/10 customers per min; 6 customers per hourRi = 1/Ta
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8Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Service Process Attributes
Ri MUST ALWAYS <= RpWe will show later that even Ri=Rp is not possibleIncoming rate must be less than processing rateThroughput = Flow Rate R = Min (Ri, Rp) Stable Process = Ri < Rp,, so that R = Ri
Safety Capacity Rs = Rp – Ri
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9Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Waiting time in the servers (processors)?Throughput?
Operational Performance Measures
Flow time T = Ti + Tp
Inventory I = Ii + Ip
Ti: waiting time in the inflow buffer
Ii: number of customers waiting in the inflow buffer
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10Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
= Utilization= inflow rate / processing rate = throughout / process capacity = R/ Rp < 1Safety Capacity = Rp – R
For example , R = 6 per hour, processing time for a single server is 6 min Rp= 12 per hour,
= R/ Rp = 6/12 = 0.5Safety Capacity = Rp – R = 12-6 = 6
Service Process Attributes
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11Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Given a single server. And a utilization of r = 0.5How many flow units are in the server ?
Operational Performance Measures
Given 2 servers. And a utilization of r = 0.5How many flow units are in the servers ?
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12Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Operational Performance Measures
Flow time T Ti + Tp
Inventory I = Ii + Ip
R
I = R T
R = I/T = Ii/Ti = Ip/Tp
Ii = R Ti Ip = R Tp
Tp if 1 server Rp = 1/Tp In general, if c servers Rp = c/Tp
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13Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Operational Performance Measures
= R/ Rp = (Ip/Tp)/(c/Tp) = Ip/c
And it is obvious that = Ip/cBecause on average there are Ip people in the
processors and the capacity of the servers is to serve c costomer at a time.
= R/Rp = Ip/c
From the Little’s Law
Rp = c/Tp
R = Ip/Tp
From definition of Rp
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14Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Sales Throughput Rate
Cost Capacity utilization Number in queue / in system
Customer service Waiting Time in queue /in system
Financial Performance Measures
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15Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
What is the queue size?What is the capacity utilization?
Arrival Rate at an Airport Security Check Point
Customer Number Arrival Time
Departure Time
Time in Process
1 0 5 5
2 4 10 6
3 8 15 7
4 12 20 8
5 16 25 9
6 20 30 10
7 24 35 11
8 28 40 12
9 32 45 13
10 36 50 14
0 10 20 30 40 50
Time
1
2
3
4
5
6
7
8
9
10
Cust
omer
Num
ber
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16Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
What is the queue size?What is the capacity utilization?
Flow Times with Arrival Every 6 Secs
Customer Number Arrival Time
Departure Time
Time in Process
1 0 5 5
2 6 11 5
3 12 17 5
4 18 23 5
5 24 29 5
6 30 35 5
7 36 41 5
8 42 47 5
9 48 53 5
10 54 59 5
0 10 20 30 40 50 60
Time
1
2
3
4
5
6
7
8
9
10
Cust
omer
Num
ber
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17Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
What is the queue size?What is the capacity utilization?
Flow Times with Arrival Every 6 Secs
Customer Number Arrival Time Processing Time
Time in Process
1-A 0 7 7
2-B 10 1 1
3-C 20 7 7
4-D 22 2 7
5-E 32 8 8
6-F 33 7 14
7-G 36 4 15
8-H 43 8 16
9-I 52 5 12
10-J 54 1 11
0 10 20 30 40 50 60 70
Time
1
2
3
4
5
6
7
8
9
10
Cu
sto
mer
Queue Fluctuation
0
1
2
3
4
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64
Time
Nu
mb
er
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18Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
What is the queue size?What is the capacity utilization?
Flow Times with Arrival Every 6 Secs
Customer Number Arrival Time Processing Time
Time in Process
1-E 0 8 8
2-H 10 8 8
3-D 20 2 2
4-A 22 7 7
5-B 32 1 1
6-J 33 1 1
7-C 36 7 7
8-F 43 7 7
9-G 52 4 4
10-I 54 5 70 10 20 30 40 50 60 70
1
2
3
4
5
6
7
8
9
10
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19Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
If inter-arrival and processing times are constant, queues will build up if and only if the arrival rate is greater than the processing rate
If there is (unsynchronized) variability in inter-arrival and/or processing times, queues will build up even if the average arrival rate is less than the average processing rate
If variability in interarrival and processing times can be synchronized (correlated), queues and waiting times will be reduced
Conclusion
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20Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Two key drivers of process performance are (i) Capacity utilization, and (ii) interarrival time and processing time variability
High capacity utilization ρ= R/ Rp or low safety capacity Rs =R – Rp, due to High inflow rate R Low processing rate Rp=c / Tp, which may be due
to small-scale c and/or slow speed 1 /Tp High, unsynchronized variability in
Interarrival times Processing times
Causes of Delays and Queues
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21Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Variability in the interarrival and processing times can be measured using standard deviation. Higher standard deviation means greater variability.
Coefficient of Variation: the ratio of the standard deviation of interarrival time (or processing time) to the mean.
Ci = coefficient of variation for interarrival times
Cp = coefficient of variation for processing times
Drivers of Process Performance
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22Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Operational Performance Measures
Flow time T = Ti + Tp
Inventory I = Ii + Ip
Ti: waiting time in the inflow buffer = ?
Ii: number of customers waiting in the inflow buffer =?
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23Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Ri / Rp, where Rp = c / Tp
Ci and Cp are the Coefficients of Variation
(Standard Deviation/Mean) of the inter-arrival and processing times (assumed independent)
The Queue Length Formula
ρ1
ρ1)2(c
Utilization effect
Variability effect
2
22pi CC
iI
![Page 24: 1 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2 Made-to-stock (MTS) operations Product is manufactured and stocked in advance of](https://reader031.vdocuments.us/reader031/viewer/2022032005/56649d405503460f94a1a372/html5/thumbnails/24.jpg)
24Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
This part factor captures the capacity utilization effect, which shows that queue length increases rapidly as the capacity utilization p increases to 1
Factors affecting Queue Length
ρ1
ρ1)2(c
iI
The second factor captures the variability effect, which shows that the queue length increases as the variability in interarrival and processing times increases.
Whenever there is variability in arrival or in processing queues will build up and customers will have to wait, even if the processing capacity is not fully utilized.
2
22pi CC
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25Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Throughput- Delay Curve
VariabilityIncreases
AverageFlowTime T
Utilization (ρ) r100%
Tp
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26Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Coefficient of Variations
Interarrival Time or Processing Time distribution General Poisson Exponential Constant
Mean Interarrival Time or Processiong Time m m m m
Standard Deviation of interarrival or Processing Time s m m 0
Coeffi cient of Varriation of Interarrival or Processing Time s/m 1 1 0
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27Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
A sample of 10 observations on Interarrival times in seconds
10,10,2,10,1,3,7,9, 2, 6=AVERAGE () Avg. interarrival time = 6Ri = 1/6 arrivals / sec.
=STDEV() Std. Deviation = 3.94Ci = 3.94/6 = 0.66
C2i = (0.66)2 = 0.4312
Example 8.4
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28Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
A sample of 10 observations on processing times in seconds
7,1,7 2,8,7,4,8,5, 1Tp= 5 secondsRp = 1/5 processes/sec.Std. Deviation = 2.83Cp = 2.83/5 = 0.57C2
p = (0.57)2 = 0.3204
Example 8.4
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29Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Ri =1/6 < RP =1/5 R = Ri
= R/ RP = (1/6)/(1/5) = 0.83
With c = 1, the average number of passengers in queue is as follows:
Example 8.4
On average 1.56 passengers waiting in line, even though safety capacity is Rs= RP - Ri = 1/5 - 1/6 = 1/30 passenger per second, or 2 per minutes
Ii = [(0.832)/(1-0.83)] ×[(0.662+0.572)/2] = 1.56
ρ1
ρ1)2(c
2
22pi CC
iI 0.831
0.831)2(1
2
57.066.0 22
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30Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Other performance measures: Ti=Ii/R = (1.56)(6) = 9.4 seconds
Compute T? T = Ti+TpSince TP= 5 T = Ti + TP = 14.4 seconds
Total number of passengers in the process is: I = RT = (1/6) (14.4) = 2.4Alternatively, 1.56 in the buffer. How many in the process?I = 1.56 + 0.83 = 2. 39
Example 8.4
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31Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Compute R, Rp and : Ta= 6 min, Tp = 5 minR = Ri= 1/6 per minuteProcessing rate for one processor 1/5 for 2
processorsRp = 2/5 = R/Rp = (1/6)/(2/5) = 5/12 = 0.417
Now suppose we have two servers.
On average 0.076 passengers waiting in line.safety capacity is Rs= RP - Ri = 2/5 - 1/6 = 7/30
passenger per second, or 14 passengers per minutes
ρ1
ρ1)2(c
2
22pi CC
iI 0.4171
0.4171)2(2
2
57.066.0 22
![Page 32: 1 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2 Made-to-stock (MTS) operations Product is manufactured and stocked in advance of](https://reader031.vdocuments.us/reader031/viewer/2022032005/56649d405503460f94a1a372/html5/thumbnails/32.jpg)
32Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Other performance measures: Ti=Ii/R = (0.076)(6) = 0.46 seconds
Compute T? T = Ti+TpSince TP= 5 T = Ti + TP = 0.46+5 = 5.46 seconds
Total number of passengers in the process is: I= 0.08 in the buffer and 0.417 in the process. I = 0.076 + 2(0.417) = 0.91
Example 8.4
c ρ Rs Ii Ti T I
1 0.83 0.03 1.56 9.38 14.38 2.4
2 0.417 0.23 0.077 0.46 5.46 0.91
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33Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Example: The arrival rate to a GAP store is 6 customers per hour and has Poisson distribution. The service time is 5 min per customer and has Exponential distribution. On average how many customers are in the waiting line? How long a customer stays in the line? How long a customer stays in the processor (with the
server)? On average how many customers are with the server? On average how many customers are in the system? On average how long a customer stay in the system?
Assignment- Problem 1: M/M/1 Performance Evaluation
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34Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Assignment- Problem 2: M/M/1 Performance Evaluation
What if the arrival rate is 11 per hour?
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35Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
A local GAP store on average has 10 customers per hour for the checkout line. The inter-arrival time follows the exponential distribution. The store has two cashiers. The service time for checkout follows a normal distribution with mean equal to 5 minutes and a standard deviation of 1 minute.
On average how many customers are in the waiting line? How long a customer stays in the line? How long a customer stays in the processors (with the
servers)? On average how many customers are with the servers? On average how many customers are in the system ? On average how long a customer stay in the system ?
Assignment- Problem 3: M/G/c
![Page 36: 1 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2 Made-to-stock (MTS) operations Product is manufactured and stocked in advance of](https://reader031.vdocuments.us/reader031/viewer/2022032005/56649d405503460f94a1a372/html5/thumbnails/36.jpg)
36Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
A call center has 11 operators. The arrival rate of calls is 200 calls per hour. Each of the operators can serve 20 customers per hour. Assume interarrival time and processing time follow Poisson and Exponential, respectively. What is the average waiting time (time before a customer’s call is answered)?
Assignment- Problem 4: M/M/c Example
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37Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines2
Suppose the service time is a constant What is the answer of the previous question?
Assignment- Problem 5: M/D/c Example