OPSM 301: Operations Management

1. Koç University OPSM 301: Operations Management Session 20: Queue Management Zeynep Aksin zaksin@ku.edu.tr

2. 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)

3. Operational Performance Measures processing waiting () Ri e.g10 /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

4. 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)

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

6. Average Time in System T Variability Increases Tp 100% r Utilization (ρ) Throughput- Delay Curve

7. 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

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

9. 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)

10. 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)

11. 2 requests/hour Accounting 4 requests/hour 2 requests/hour 4 requests/hour Finance 3 requests/hour 4 requests/hour Marketing 2 requests/hour 4 requests/hour Operations Current Situation

12. Current Situation: waiting times Accounting, Operations, Finance: T =processing time+waiting time =0.25 hrs. +0.25 hrs =0.5 hrs=30 min Marketing: T =processing time+waiting time =0.25 hrs. +0.75 hrs =1 hr=60 min

13. 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

14. Proposed System: Secreterial pool 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

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

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

17. 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

18. 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.

19. 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?

20. 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= 5seconds, so • T = 14.38 seconds, so • I = RT = 14.38/6 = 2.3966 customers in the system

21. 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= 5seconds, so • T = 5.45216 seconds, so • I = RT = 5.45216/6 = 0.9087 customers in the system

22. 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?

23. 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

24. 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)