Managing Waiting Time in Service Operations: An Introduction to Queuing Theory

1. Koç University OPSM 405 Service Management Class 19: Managing waiting time: Queuing Theory Zeynep Aksin zaksin@ku.edu.tr

2. it takes 8 minutes to serve a customer 6 customers call per hour one customer every 10 minutes Flow Time = 8 min 100% 100% 90% 90% 80% 80% 70% 70% 60% 60% 50% 50% 40% 40% 30% 30% 20% 20% 10% 10% 0% 0% 0 15 30 45 60 75 90 105 120 135 150 165 180 195 Telemarketing: deterministic analysis Flow Time Distribution Probability Flow Time (minutes)

3. In reality service times exhibit variability In reality arrival times exhibit variability 25% 100% 90% 20% 80% 15% 60% Probability 10% 40% 5% 20% 0% 0% 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 More Flow Time 30% 100% 90% 25% 80% 70% 20% 60% Probability 15% 50% 40% 10% 30% 20% 5% 10% 0% 0% 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 More Flow Time Telemarketing with variability in arrival times + activity times

4. Call # 10 9 8 7 6 5 4 3 2 1 0 0 20 40 60 80 1 0 0 TIME Inventory (# of calls in system) 5 4 3 2 1 0 0 20 40 60 80 1 0 0 TIME Why do queues form? • utilization: • throughput/capacity • variability: • arrival times • service times • processor availability

5. A measure of variability • Needs to be unitless • Only variance is not enough • Use the coefficient of variation • CV= s/m

6. Interpreting the variability measures CVi = coefficient of variation of interarrival times i) constant or deterministic arrivals CVi = 0 ii) completely random or independent arrivals CVi =1 iii) scheduled or negatively correlated arrivals CVi < 1 iv) bursty or positively correlated arrivals CVi > 1

7. Little’s Law Inventory I [units] ... ... ... ... ... Flow Time T[hrs] or WIP = THROUGHPUT RATE x FLOWTIME For a queue: N=l W

8. A Queueing System c m, CVs Arrival Departure l, CVa nL tL nS tS

9. What to manage in such a process? • Inputs • Arrival rate / distribution • Service or processing time / distribution • System structure • Number of servers c • Number of queues • Maximum queue capacity/buffer capacity K • Operating control policies • Queue-service discipline

10. Performance Measures • Sales • Throughput • Abandoning rate • Cost • Capacity utilization • Queue length / total number in process • Customer service • Waiting time in queue / total time in process • Probability of blocking

11. The A/B/C notation • A: type of distribution for interarrival times • B: type of distribution for service times • C: the number of parallel servers M = exponential interarrival and service time distribution (same as Poisson arrival or service rate) D= deterministic interarrival or service time G= general distributions

12. Variation characteristics • distribution type M: CVa= CVs =1 • distribution type D: CVa= CVs = 0 • distribution type G: could be any value

13. Basic notation l = mean arrival rate (units per time period) m = mean service rate (units per time period) r = l/m = utilization rate (traffic intensity) c = number of servers (sometimes also s) P0 = probability that there are 0 customers in the system Pn = probability that there are n customers in the system Ls = mean number of customers in the system (Ns) Lq = mean number of customers in the queue (Nq) Ws = mean time in the system Wq = mean time in the queue

14. Recall Little’s Law Lq = l Wq queue length = arrival rate * time in queue

15. The building block: M/M/1 • An infinite or large population of customers arriving independently; no reservations • Poisson arrival rate (exponential interarrival times) • single server, single queue • no reneges or balking • no restrictions on queue length • first-come first-served (FCFS) • exponential service times

16. Facts for M/M/1 r < 1 P0 = 1-r Pn = P0rn Ls = l /(m-l) Ws = 1 / (m-l) Lq = r l / (m-l) Wq = r 1 / (m-l)

17. For a general system with c servers W (or tS) = average service time + Wq (or tq ) Average wait = (scale effect) (utilization effect) (variability effect) Wq = Lq / l r=l/cm Note:

18. Average Flow Time Ws Variability Increases 1/m 100% r Utilization (ρ) Generalized Throughput-Delay Curve

19. 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

20. Levers to reduce waiting and increase QoS: variability reduction + safety capacity • How to reduce system variability? • Safety Capacity = capacity carried in excess of expected demand to cover for system variability • it provides a safety net against higher than expected arrivals or services and reduces waiting time

21. Excel does it all!

22. Example: Secretarial Pool • 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

23. Current Situation 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

24. Current Situation: queueing notation l = 2 requests/hour m= 4 requests/hour Acc., Fin., Ops. C2[A] = 1 (totally random arrivals) C2[S] = 1 (assumption) l = 3 requests/hour m = 4 requests/hour Marketing C2[A] = 1 (totally random arrivals) C2[S] = 1 (assumption)

25. Current Situation: waiting times Accounting, Operations, Finance: W = service time + Wq W = 0.25 hrs. + 0.25 hrs = 30 minutes Marketing: W = service time + Wq W = 0.25 hrs. + 0.75 hrs = 60 minutes

26. Proposal: Secretarial Pool Accounting 2 Finance 2 16 requests/hour 3 Marketing 9 requests/hour 2 Operations

27. Proposal: Secretarial Pool Wq = 0.0411 hrs. W= 0.0411 hrs. + 0.25 hrs.= 17 minutes 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)

28. 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

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

30. Capacity design using queueing models • Criteria for design • waiting time • probability of excessive waiting • minimize probability of lost sales • maximize revenues

31. Example: bank branch • 48 customers arrive per hour, 50 % for teller service and 50 % for ATM service • On average, 5 minutes to service each request or 12 per hour. • Can model as two independent queues in parallel, each with mean arrival rate of l=24 customers per hour • Want to find number of tellers and ATMs to ensure customers will find an available teller or ATM at least 95 % of the time

32. How many tellers and ATMs? P(delay) or P(wait) less than 5%: 6 Tellers and 6 ATMs

33. Example • A mail order company has one department for taking customer orders and another for handling complaints. Currently each has a separate phone number. Each department has 7 phone lines. Calls arrive at an average rate of 1 per minute and are served at 1.5 per minute. Management is thinking of combining the departments into a single one with a single phone number and 14 phone lines. • The proportion of callers getting a busy signal will….? • Average flow experienced by customers will….?

34. Example • 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 YTL 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 YTL 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?

35. Example Global airlines is revamping its check-in operations at its hub terminal. This is a single queue system where an available server takes the next passenger. Arrival rate is estimated to be 52 passengers per hour. During the check-in process, an agent confirms reservation, assigns a seat, issues a boarding pass, and weighs, labels, dispatches baggage. The entire process takes on average 3 minutes. Agents are paid 20 YTL an hour and it is estimated that Global loses 1 YTL for every minute a passenger spends waiting in line. How many agents should Global staff at its hub terminal? How many agents does it need to meet the industry norm of 3 minutes wait?

36. Capacity Management • First check if average capacity is enough: is there a perpetual queue? If not, increase capacity • Capacity may be enough on average but badly distributed over time periods experiencing demand fluctuations: check if there is a predictable queue, do proper scheduling; you may need more people to accommodate scheduling constraints • Find sources of variability and try to reduce them: these create the stochastic queue

37. 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

38. Tips for queueing problems • Make sure you use rates not times for l and m • Use consistent units: minutes, hours, etc. • If the problem states “constant service times” or an “automated machine with practically constant times” this means: deterministic service so CVs=0 • Check the objective: • Cost minimization? • Service level satisfaction at lowest cost? • Etc. • Read carefully to understand difference between “waiting”, “standing in line” (in queue)“in system” or “total flow time” or “providing service”