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Waiting Times

Waiting Times. Chapter 6. Learning Objectives. Interarrival and Service Times and their variability Obtaining the average time spent in the queue Pooling of server capacities Priority rules. Where are the queues?. Americans spend > 100 M hours/day waiting in a line.

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Waiting Times

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  1. Waiting Times Chapter 6

  2. Learning Objectives • Interarrival and Service Times and their variability • Obtaining the average time spent in the queue • Pooling of server capacities • Priority rules

  3. Where are the queues? Americans spend > 100 M hours/day waiting in a line. T. Heymann in his book “On an average day”

  4. A Queue is made of a server and a queue in front Processing Buffer Input: Passengers in an airport Customers at a bank Patients at a hospital Callers at a call center Arrival rate Resources: Check-in clerks at an airport Tellers at a bank Nurses at a hospital Customer service representatives (CRS) at a call center Capacity Queues We are interested in the waiting times in the queue and the queue length.

  5. An Example of a Simple Queuing System Answered Calls Incoming calls Reps processing calls Calls on Hold • At peak, 80% of calls dialed received a busy signal. • Customers getting through had to wait on average 10 minutes • Extra phone line expense per day for waiting was $25,000. Call center Blocked calls (busy signal) Abandoned calls (tired of waiting) Financial consequences Lost throughput Holding cost Lost goodwill Lost throughput (abandoned) Cost of capacity Cost per customer $$$ Revenue $$$

  6. A Somewhat Odd Service ProcessConstant Arrival Rate (0.2/min) and Service Times (4 min) Arrival Time Service Time Patient 1 0 4 2 5 4 3 10 4 4 15 4 5 20 4 6 25 4 7 30 4 8 35 4 9 40 4 10 45 4 11 50 4 12 55 4 7:00 7:10 7:20 7:30 7:40 7:50 8:00

  7. Where is Variability? • There certainly is significant (actually infinite) amount waiting when the arrival rate is greater than the service rate • Equivalently, the processing capacity is less than the flow rate • More interestingly, variability can cause long waiting times. Variability in • Arrival process • Processing times • Availability of resources • Routing of flow units; Recall the Resume Validation Example.

  8. Variability: Where does it come from? Examples • Tasks: • Inherent variation • Lack of Standard Operating Procedures • Quality (scrap / rework) Processing Buffer • Input: • Unpredicted Volume swings • Random arrivals (randomness is the rule, not the exception) • Incoming quality • Product Mix • Resources: • Breakdowns / Maintenance • Operator absence • Set-up times • Especially relevant in service operations • (what is different in service industries?): • emergency room • air-line check in • call center • check-outs at cashier • Routes: • Variable routing • Dedicated machines

  9. A More Realistic Service ProcessRandomArrival Rate and Service Times Arrival Time Service Time Patient 1 0 5 Patient 1 Patient 3 Patient 5 Patient 7 Patient 9 Patient 11 2 7 6 3 9 7 Patient 2 Patient 4 Patient 6 Patient 8 Patient 10 Patient 12 4 12 6 5 18 5 6 22 2 7 25 4 Time 8 30 3 9 36 4 7:00 7:10 7:20 7:30 7:40 7:50 8:00 10 45 2 11 51 2 12 55 3 3 2 Number of cases 1 0 2 min. 3 min. 4 min. 5 min. 6 min. 7 min. Service times

  10. Variability Leads to Waiting TimeAverage Arrival Rate (0.2/min) and Service Times (4 min) Arrival Time Service Time Patient 1 0 5 2 7 6 3 9 7 4 12 6 5 18 5 6 22 2 7 25 4 8 30 3 9 36 4 10 45 2 11 51 2 12 55 3 Service time Wait time 5 4 3 2 1 Inventory (Patients at lab) 0 7:00 7:10 7:20 7:30 7:40 7:50 8:00

  11. 160 160 140 140 120 120 100 100 80 80 60 60 40 40 20 20 0 0 Time Time Is the incoming call rate stationary? Number of customers Per 15 minutes 12:30 10:45 23:00 21:15 17:45 19:30 14:15 16:00 2:00 3:45 7:15 0:15 9:00 5:30

  12. Cumulative Number of Customers Cumulative Number of Customers 70 700 60 600 50 500 Expected arrivalsif stationary 40 400 30 300 20 200 Actual, cumulative arrivals 10 100 0 0 0 0 6:00:00 6:00:00 6:00:00 7:00:00 7:00:00 7:00:00 8:00:00 8:00:00 8:00:00 9:00:00 9:00:00 9:00:00 10:00:00 10:00:00 10:00:00 7:15:00 7:15:00 7:18:00 7:18:00 7:21:00 7:21:00 7:24:00 7:24:00 7:27:00 7:27:00 7:30:00 7:30:00 Time Time How to test for stationary?

  13. Exponential distribution for Interarrival times 100 1 90 80 0.8 70 Probability{Interarrival time  t} Number of calls with given duration t 60 0.6 50 40 0.4 30 20 0.2 10 0 0 time 0 1 2 3 0.8 1.2 1.6 2.8 3.2 0.2 0.4 0.6 1.4 1.8 2.2 2.4 2.6 Duration t Prob { IA <=t } = 1-exp(-t/a) E(IA)=a, expected time between arrivals StDev(IA)=a

  14. Comparing empirical and theoretical distributions Distribution Function 1 1 Exponential distribution 0.8 0.8 0.6 0.6 Empirical distribution 0.4 0.4 0.2 0.2 - - 0 0 Interarrival time 0:00:00 0:00:00 0:00:09 0:00:09 0:00:17 0:00:17 0:00:26 0:00:26 0:00:35 0:00:35 0:00:43 0:00:43 0:00:52 0:00:52 0:01:00 0:01:00 0:01:09 0:01:09

  15. Analyzing the Arrival Process • CVa=StDev(IA)/E(IA) Stationary Arrivals? YES NO Exponentially distributed inter-arrival times? Break arrival process up into smaller time intervals YES NO • Compute a: average interarrival time • CVa=1 • All results of chapters 6 and 7 apply • Compute a: average interarrival time • CVa= St.dev. of interarrival times / a • Results of chapter 6 or 7 do not apply, require simulation or more complicated models

  16. 800 Frequency 600 400 Call duration[minutes] 2.5 2 1.5 1 0.5 0 Week-end averages 200 Std. Dev = 141.46 Call durations [seconds] Mean = 127.2 N = 2061.00 0 0 300 600 100 400 800 900 200 500 700 1000 Week-day averages A Time of the day 0:00 23:00 Analyzing the Service TimesSeasonality and Variability CVp: Coefficient of variation of service times

  17. Inventory waiting Iq Inventory in service Ip Inflow Outflow Entry to system Begin Service Departure Waiting Time Tq Service Time p Flow Time T=Tq+p Computing the expected waiting time Tq

  18. Utilization Example: Average Activity time=p=90 seconds Average Interarrival time=a=300 seconds Utilization=90/300=0.3=30%

  19. The Waiting Time Formula Waiting Time Formula for Exponential Arrivals Variability factor Utilization factor Service time factor Example: Average Activity time=p=90 seconds Average Interarrival time=a=300 seconds CVa=1 and CVp=1.333

  20. Bank Teller Example • An average of 10 customers per hour come to a bank teller who serves each customer in 4 minutes on average. Assume exponentially distributed interarrival and service times. • What is the teller’s utilization? • What is the average time spent in the queue waiting? • How many customers would be waiting for this teller or would be serviced by this teller on average? • On average, how many customers are served per hour? Answer: p=4 mins; a=6 mins=60/10

  21. The Flow Time Increase Exponentially in Utilization Average flow time T Increasing Variability Theoretical Flow Time Utilization 100%

  22. Computing Tq with m Parallel Servers Inventory in the system I=Iq+ Ip Inventory in service Ip Inventory waiting Iq Outflow Inflow Entry to system Begin Service Departure Waiting Time Tq Service Time p Flow Time T=Tq+p

  23. Utilization with m servers Example: Average Activity time=p=90 seconds Average Interarrival time=a=11.39 secs over 8-8:15 m=10 servers Utilization=90/(10x11.39)=0.79=79%

  24. Waiting Time Formula for Parallel Resources Approximate Waiting Time Formula for Multiple (m) Servers Example: Average Activity time=p=90 seconds Average Interarrival time=a=11.39 seconds m=10 servers CVa=1 and CVp=1.333

  25. Online Retailer Example • Customers send emails to a help desk of an online retailer every 2 minutes, on average, and the standard deviation of the inter-arrival time is also 2 minutes. The online retailer has three employees answering emails. It takes on average 4 minutes to write a response email. The standard deviation of the service times is 2 minutes. • Estimate the average customer wait before being served. • How many emails would there be -- on average -- that have been submitted to the online retailer, but not yet answered? • Answer: a=2 mins; CVa=1; m=3; p=4 mins; CVp=0.5 • a) Find Tq. b) Find Iq=(1/a)Tq

  26. Service Levels in Waiting Systems 90% of calls had to wait 25 seconds or less Fraction of customers who have to wait xseconds or less 1 Waiting times for those customers who do not get served immediately 0.8 0.6 Fraction of customers who get served without waiting at all 0.4 0.2 0 0 50 100 150 200 Waiting time [seconds] • Target Wait Time (TWT) • Service Level = Probability{Waiting TimeTWT}; needs distribution of waiting time • Example: Deutsche Bundesbahn Call Center - now (2003): 30% of calls answered within 20 seconds - target: 80% of calls answered within 20 seconds

  27. Bank of America’s Service Measures

  28. Waiting Lines: Points to Remember • Variability is the norm, not the exception - understand where it comes from and eliminate what you can - accommodate the rest • Variability leads to waiting times although utilization<100% • Use the Waiting Time Formula to - get a qualitative feeling of the system - analyze specific recommendations / scenarios • Adding capacity is expensive, although some safety capacity is necessary • Next case: - application to call center - use CV=1 - careful in interpreting March / April call volume

  29. Summary of the formulas

  30. Cost of direct labor per serviced unit Ex: $10/hour wage for each CSR; m=10 Activity time=p=90 secs; Interarrival time=11.39 secs 1-800 number line charge $0.05 per minute

  31. Cost of Staffing Levels The optimal staffing level m=10

  32. Number ofCSRs Number of customers Number of customers Per 15 minutes Per 15 minutes 160 160 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 140 140 120 120 100 100 80 80 60 60 40 40 20 20 0 0 Time Time 0:15 0:15 2:00 2:00 3:45 3:45 5:30 5:30 7:15 7:15 9:00 9:00 21:15 21:15 23:00 23:00 10:45 10:45 12:30 12:30 14:15 14:15 16:00 16:00 17:45 17:45 19:30 19:30 Staffing Levels under various Interarrival Times

  33. The Power of Pooling Independent Resources 2x(m=1) Waiting Time Tq 70.00 m=1 60.00 50.00 40.00 m=2 30.00 20.00 m=5 Pooled Resources (m=2) 10.00 m=10 0.00 60% 65% 70% 75% 80% 85% 90% 95% Utilization u Implications:+ balanced utilization + Shorter waiting time (pooled safety capacity) - Change-overs / set-ups

  34. Service-Time-Dependent Priority Rules A A C C B B D D 4 min. 9 min. 19 min. 12 min. 23 min. 21 min. Total wait time: 9+19+23=51min Total wait time: 4+12+21=37 min • Flow units are sequenced in the waiting area (triage step) • Shortest Processing Time (SPT) Rule - Minimizes average waiting time • Do you want to wait for a short process or a long one? Service times: A: 9 minutesB: 10 minutesC: 4 minutesD: 8 minutes • Problem of having “true” processing times

  35. Service-Time-Independent Priority Rules W(t) Time t • Sequence based on importance - emergency cases; identifying profitable flow units • First-Come-First-Serve - easy to implement; perceived fairness • The order in which customers are served does Not affect the average waiting time. • W(t): Work in the system • An arrival at t waits until the work W(t) is completed p2 p1 First arrival Second arrival Last departure

  36. Service-Time-Independent Order does not affect the waiting time Order: 1-1-2 Order: 1-2-1 W(t) W(t) Time t Time t Secondfinishes Firstfinishes Work is conserved even when the processing order changes. No matter what the order is, the third arrival finds the same amount of work W(t).

  37. Operator Performance in a Call Center Call durations 2:00 min. Operator KB short Operator NN 2:30 min. Operator BJ 3:00 min. Operator BK 3:30 min. Operator NJ 4:00 min. long 4:30 min. Low courtesy High courtesy Courtesy / Friendliness (qualitative information) Also has highest variability

  38. Summary • Interarrival and Service Times and their variability • Obtaining the average time spent in the queue • Pooling of server capacities • Priority rules

  39. HW question: Waiting time formula • Consider the waiting time formula in a queue: • In this exercise, we shall simplify this formula and write it in a different form when the interarrival times are exponentially distributed. • a) Use the coefficient of variation (CV) of exponential distribution to simplify the formula above. • b) By expressing the utilization in terms of “expected activity time p” and “expected interarrival time a”, put the simplified formula in part a) into the following form: • Find the term in square brackets above and express it in English. • c) What does the term in square brackets reduce to if the activity time is constant and is p.

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