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OPSM 301: Operations Management Session 20: Queue Management Koç University Zeynep Aksin [email protected]
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)
Operational Performance Measures
Flow time T = Tw + Tp (waiting+process)Inventory I = Iw + Ip
Flow Rate R = Min (Ri, RpStable 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
waiting processing() Ri
e.g10 /hr
R ()
10 /hr
10 min, Rp=12/hrTw?
Summary: Causes of Delays and Queues High Unsynchronized Variability in
– Interarrival Times– Processing Times
High Capacity Utilization = 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)
The Queue Length Formula
2ρ1
221)2(cρ pi
w
CVCVI
Utilization effect
Variability effectx
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)
VariabilityIncreases
AverageTime inSystem T
Utilization (ρ) 100%
Tp
Throughput- Delay Curve
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
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
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)
2ρ1
221)2(cρ pi
w
CVCVI
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)
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Current SituationCurrent Situation
Accounting
Finance
Marketing
Operations
2 requests/hour
2 requests/hour
3 requests/hour
2 requests/hour
4 requests/hour
4 requests/hour
4 requests/hour
4 requests/hour
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Current Situation: waiting times
T =processing time+waiting time =0.25 hrs. + 0.25 hrs =0.5 hrs=30 min
Accounting, Operations, Finance:
Marketing:
75.0325.2
25.275.01
75.02
110.75-1
75.0
1c ,75.043
22x2
w
w
T
I
25.025.0
5.05.01
5.02
110.5-1
5.0
1c ,5.042
22x2
RIT
I
ww
w
T =processing time+waiting time =0.25 hrs. + 0.75 hrs =1 hr=60 min
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Proposal: Secretarial PoolProposal: Secretarial PoolAccounting
Finance
Marketing
Operations
9 requests/hour
22
3
2
Arrival rate=R=9/hr Tp=1/4 hr, Rp=c/Tp=16/hr
Utilization=Ri/Rp=9/16
Proposed System: Secreterial pool
04.0937.0
37.02
115625.01
5625.0
9R 4 5625.0169
)14(2
RIT
I
c
ww
w
T =processing time+waiting time =0.25 hrs. + 0.04 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
Server 1Queue 1
Server 2Queue 2
Server 1
Queue
Server 2
Effect of Pooling
Ri
Ri
Ri/2
Ri/2
Pooled service capacityreduces waiting
Examples of pooling in business
Consolidating back office work Call centers Single line versus separate queues
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
Intuition building exercise
Check the following website: Waiting Line Simulation (use internet explorer)
http://archive.ite.journal.informs.org/Vol7No1/DobsonShumsky/security_simulation.php
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.
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Exercise: Example 1
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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=0.66CVi=1, CVp=0
min84.7
064.01066.0
customers 64.021
66.0166.0 2
pw
ww
w
TTT
hrRIT
I
Exercise: 1. What if we have a human server, with CV=1?2.What is the effect of buying a second machine?
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 = 0.6562– CVp = 2.8284/5 = 0.5657
Queue Length Formula– Iw = 1.5633
Hence– Tw = Iw / R = 9.38 seconds, and Tp = 5 seconds, so – T = 14.38 seconds, so– I = RT = 14.38/6 = 2.3966 customers in the system
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 = 0.4167– CVi = 3.937/6 = 0.6562– CVp = 2.8284/5 = 0.5657
Queue Length Formula– Iw = 0.07536
Hence– Tw = Iw / R = 0.45216 seconds, and Tp = 5 seconds, so – T = 5.45216 seconds, so– I = RT = 5.45216/6 = 0.9087 customers in the system
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.a. 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?
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
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)
