1. Managing Flow Variability: Safety Capacity
OPSM301 Spring 2012
Class 19:
Managing Flow Variability: Safety Capacity or
Management of Queues

2. Economics of the Waiting Line Problem
A central problem in many service settings is the management of waiting time
Reducing waiting time costs money
Waiting time also costs money
When people waiting are customers, it is difficult to value their time
Lost sales is one value

3. Components of the Queuing System Visually
Customers come in
Customers leave
Customers are served

4. The Service Process Customer Inflow (Arrival) Rate (Ri) ()
Inter-arrival Time = 1 / Ri
Processing Time Tp (unit load)
Processing Rate per Server = 1/ Tp (µ)
Number of Servers (c)
Number of customers that can be processed simultaneously
Total Processing Rate (Capacity) = Rp= c / Tp (cµ)

5. Operational Performance Measures
processing
waiting
() Ri
e.g 10 /hr
R ()
10 /hr
Tw?
10 min, Rp=12/hr
Flow time T = Tw + Tp (waiting+process)
Inventory I = Iw + Ip
Flow Rate R = Min (Ri, Rp)
Stable Process = Ri < Rp,, so that R = Ri
Little’s Law: I = R  T, Iw = R  Tw, Ip = R  Tp
Capacity Utilization = Ri / Rp < 1
Safety Capacity = Rp – Ri
Number of Busy Servers = Ip= c = Ri  Tp

6. Flow Times with Arrival Every 4 Secs (Service time=5 seconds)
Customer Number
Arrival Time
Departure Time
Time in Process 1 5 2 4 10 6 3 8 15 7 12 20 16 25 9 30 24 35 11 28 40 32 45 13 36 50 14
What is the queue size? Can we apply Little’s Law?
What is the capacity utilization?

7. Flow Times with Arrival Every 6 Secs (Service time=5 seconds)
Customer Number
Arrival Time
Departure Time
Time in Process 1 5 2 6 11 3 12 17 4 18 23 24 29 30 35 7 36 41 8 42 47 9 48 53 10 54 59
What is the queue size?
What is the capacity utilization?

8. Effect of Variability What is the queue size?
Customer Number
Arrival Time
Processing Time
Time in Process 1 7 2 10 3 20 4 22 5 32 8 6 33 14 36 15 43 16 9 52 12 54 11
Average=6
Average=5
What is the queue size?
What is the capacity utilization?

9. Effect of Synchronization
Customer Number
Arrival Time
Processing Time
Time in Process 1 8 2 10 3 20 4 22 7 5 32 6 33 36 43 9 52 54
What is the queue size?
What is the capacity utilization?

10. Conclusion
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 inter-arrival and processing times can be synchronized (correlated), queues and waiting times will be reduced

11. Summary: Causes of Delays and Queues
High Unsynchronized Variability in
Interarrival Times
Processing Times
High Capacity Utilization r = Ri / Rp, or Low Safety Capacity Rs = Rp – Ri, due to
High Inflow Rate Ri
Low Processing Rate Rp = c/ Tp (i.e. long service time, or few servers)

12. The Queue Length Formula
Utilization effect
Variability effect x
where  Ri / Rp, where Rp = c / Tp, and
CVi and CVp are the Coefficients of Variation
(Standard Deviation/Mean) of the inter-arrival and processing times (assumed independent)

13. Throughput- Delay Curve
Variability
Increases
Average
Time in
System T
Utilization (ρ) r
100% Tp
We must have
slack capacity
ρ < 1

14. In words:
In high utilization case: small decrease in utilization yields large improvement in response time
This marginal improvement decreases as the slack in the system increases

15. Deriving Performance Measures from Queue Length Formula
Use the formula to find Iw
Tw = Iw /R
T = Tw + Tp
Ip = Tp R
I =Iw + Ip

16. How can we reduce waiting?
Reduce utilization:
Increase capacity: faster servers, better process design, more servers
Reduce variability
Arrival: Appointment system
Service:Standardization of processes, automation
We can control arrivals
Short lines (express cashiers)
Specific hours for specific customers
Specials (happy hour)

17. Example :Effect of pooling
4 Departments and 4 Departmental secretaries
Request rate for Operations, Accounting, and Finance is 2 requests/hour
Request rate for Marketing is 3 requests/hour
Secretaries can handle 4 requests per hour
Marketing department is complaining about the response time of the secretaries. They demand 30 min. response time
College is considering two options:
Hire a new secretary
Reorganize the secretarial support
Assume inter-arrival time for requests and service times have exponential distribution (i.e. CV=1)

18. Current Situation 2 requests/hour Accounting 4 requests/hour Finance
Marketing
Operations
2 requests/hour
3 requests/hour
4 requests/hour

19. Current Situation: waiting times
Accounting, Operations, Finance:
T =processing time+waiting time
=0.25 hrs hrs =0.5 hrs=30 min
Marketing:
T =processing time+waiting time
=0.25 hrs hrs =1 hr=60 min

20. Proposal: Secretarial Pool
Accounting 2
Finance 2 3
Marketing
9 requests/hour 2
Operations
Arrival rate=R=9/hr
Tp=1/4 hr, Rp=c/Tp=16/hr
Utilization=Ri/Rp=9/16

21. Proposed System: Secreterial pool
T =processing time+waiting time
=0.25 hrs hrs =0.29 hr=17.4 min
In the proposed system, faculty members in all departments
get their requests back in 17 minutes on the average. (Around 50% improvement for Acc, Fin, and Ops and 75% improvement for Marketing).
Pooling improves waiting times by ensuring effective use of capacity

22. Effect of Pooling Pooled service capacity reduces waiting Ri/2
Server 1
Queue 1
Server 2
Queue 2
Ri/2 Ri
Ri/2
Server 1
Pooled service capacity
reduces waiting Ri
Queue
Server 2

23. Examples of pooling in business
Consolidating back office work
Call centers
Single line versus separate queues

24. The impact of task integration (pooling)
balances utilization...
reduces resource interference...
...therefore reduces the impact of temporary bottlenecks
there is more benefit from pooling in a high utilization and high variability process
pooling is beneficial as long as
it does not introduce excessive variability in a low variability system
the benefits exceed the task time reductions due to specialization

25. Intuition building exercise
Check the following website:
Waiting Line Simulation (use internet explorer)
Run six different examples. Suggestion (you can use different numbers):
Arrival rate=9, service rate=10 , CV=0, CV=1, CV=2 CV=0.5
Arrival rate =9, service rate=12 CV=1 CV=0.5
Write down the parameters and the average performance measures to observe the effect of utilization and variability on waiting times. Compare the simulation output with the results you find using formulas. Note the effect of variability and utilization.

26. Exercise: Example 1
An automated pizza vending machine heats and dispenses a slice of pizza in exactly 4 minutes. Customers arrive at a rate of one every 6 minutes with the arrival rate exhibiting a Poisson distribution.
Determine:
A) The average number of customers in line.
B) The average total waiting time in the system.
Ri=1/6 per min=10/hr
Tp=4 min, c=1 Rp =15/hr =10/15=0.66
CVi=1, CVp=0
Exercise: 1. What if we have a human server, with CV=1?
2.What is the effect of buying a second machine?

27. Exercise Example 2: Computing Performance Measures
Given
Interarrival times: 10, 10, 2, 10, 1, 3, 7, 9, and 2 seconds
Avg=6, stdev=3.937, Ri =1/6
Processing times: 7, 1, 7, 2, 8, 7, 4, 8, 5, 1 seconds
Avg=5, stdev=2.8284
c = 1, Rp =1/5
Compute
Capacity Utilization r = Ri / Rp = 5/6=0.833
CVi = 3.937/6 =
CVp = /5 =
Queue Length Formula
Iw =
Hence
Tw = Iw / R = 9.38 seconds, and Tp = 5 seconds, so
T = seconds, so
I = RT = 14.38/6 = customers in the system

28. Example 2:Effect of Increasing Capacity
Assume an indentical server is added (c=2). Given
Interarrival times: 10, 10, 2, 10, 1, 3, 7, 9, and 2
Avg=6, stdev=3.937, Ri =1/6
Processing times: 7, 1, 7, 2, 8, 7, 4, 8, 5, 1
Avg=5, stdev=2.8284
c = 2, Rp =2/5
Compute
Capacity Utilization r = Ri / Rp =
CVi = 3.937/6 =
CVp = /5 =
Queue Length Formula
Iw =
Hence
Tw = Iw / R = seconds, and Tp = 5 seconds, so
T = seconds, so
I = RT = /6 = customers in the system

29. Exercise: Capacity planning
A bank would like to improve its drive-in service by reducing waiting and transaction times. Average rate of customer arrivals is 30/hour. Customers form a single queue and are served by 4 windows in a FCFS manner. Each transaction is completed in 6 minutes on average. The bank is considering to lease a high speed information retrieval and communication equipment that would cost 30 TL per hour. The facility would reduce each teller’s transaction time to 4 minutes per customer.
If our manager estimates customer cost of waiting in queue to be 20 TL per customer per hour, can she justify leasing this equipment?
b. The competitor provides service in 8 minutes on average. If the bank wants to meet this standard, should it lease the new equipment?

30. Want to eliminate as much variability as possible from your processes: how?
specialization in tasks can reduce task time variability
standardization of offer can reduce job type variability
automation of certain tasks
IT support: templates, prompts, etc.
Incentives
Scheduled arrivals to reduce demand variability
Initiatives to smoothen arrivals

31. Want to reduce resource interference in your processes: how?
smaller lotsizes (smaller batches)
better balanced line
by speeding-up bottleneck (adding staff, changing procedure, different incentives, change technology)
through cross-training
eliminate steps
buffers
integrate work (pooling)