Waiting Times Chapter 7 1
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? 3
A Queue is made of a server and a queue in front Buffer Processing Input: Passengers in an airport Customers at a bank Patients at a hospital Callers at a call center Arrival rate Queues Resources: Check-in clerks at an airport Tellers at a bank Nurses at a hospital Customer service representatives (CRS) at a call center Capacity We are interested in the waiting times in the queue and the queue length.
An Example of a Simple Queuing System Incoming calls Calls on Hold Reps processing calls Answered Calls Blocked calls (busy signal) Abandoned calls (tired of waiting) Call center 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 $5,000. Financial consequences Lost throughput Holding cost Lost goodwill Lost throughput (abandoned) Cost of capacity Cost per customer $$$ Revenue $$$ 5
A Somewhat Odd Service Process Constant Arrival Rate (0./min) and Service Times ( min) Patient 1 3 5 6 7 8 9 10 11 1 Arrival Time 0 5 10 15 0 5 30 35 0 5 50 55 Service Time 7:00 7:10 7:0 7:30 7:0 7:50 8:00 6
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. 7
Variability: Where does it come from? Examples Tasks: Inherent variation Lack of Standard Operating Procedures Quality (scrap / rework) Input: Unpredicted Volume swings Random arrivals (randomness is the rule, not the exception) Incoming quality Product Mix Especially relevant in service operations (what is different in service industries?): emergency room air-line check in call center check-outs at cashier Buffer Processing Resources: Breakdowns / Maintenance Operator absence Set-up times Routes: Variable routing Dedicated machines
A More Realistic Service Process Random Arrival Rate and Service Times Patient 1 Arrival Time 0 Service Time 5 Patient 1 Patient 3 Patient 5 Patient 7 Patient 9 Patient 11 Patient Patient Patient 6 Patient 8 Patient 10 Patient 1 7 6 3 9 7 Time 1 6 7:00 7:10 7:0 7:30 7:0 7:50 8:00 5 18 5 6 7 5 3 8 9 10 11 30 36 5 51 3 Number of cases 1 1 55 3 0 min. 3 min. min. 5 min. 6 min. 7 min. Service times 9
Variability Leads to Waiting Time Average Arrival Rate (0./min) and Service Times ( min) Patient 1 Arrival Time 0 Service Time 5 Service time 7 6 3 9 7 1 6 5 6 18 5 Wait time 7 5 8 30 3 9 36 5 10 11 1 5 51 55 3 3 Inventory (Patients at lab) 1 0 7:00 7:10 7:0 7:30 7:0 7:50 8:00 10
Is the incoming call rate stationary? Number of customers Per 15 minutes 160 10 10 100 80 60 0 0 0 0:15 :00 3:5 5:30 7:15 9:00 10:5 1:30 1:15 16:00 17:5 19:30 1:15 3:00 Time 11
How to test for stationary? Cumulative Number of Customers Cumulative Number of Customers 700 70 600 60 500 00 Expected arrivals if stationary 50 0 300 30 00 100 Actual, cumulative arrivals 0 10 0 0 6:00:00 7:00:00 8:00:00 9:00:00 10:00:00 Time 7:15:00 7:18:00 7:1:00 7::00 7:7:00 7:30:00 Time 1
Exponential distribution for Interarrival times Number of calls with given duration t 100 90 80 70 60 50 0 30 0 10 Probability{Interarrival time t} 1 0.8 0.6 0. 0. 0 0 time Duration t Prob { IA <=t } = 1-exp(-t/a) E(IA)=a, expected time between arrivals StDev(IA)=a 13
Comparing empirical and theoretical distributions Distribution Function 1 0.8 Exponential distribution 0.6 0. Empirical distribution 0. 0 0:00:00 0:00:09 0:00:17 0:00:6 0:00:35 0:00:3 0:00:5 0:01:00 0:01:09 - Interarrival time 1
Analyzing the Arrival Process CV a =StDev(IA)/E(IA) Stationary Arrivals? YES NO YES Exponentially distributed inter-arrival times? NO Break arrival process up into smaller time intervals Compute a: average interarrival time CV a =1 All results of chapters 7 and 8 apply Compute a: average interarrival time CV a = St.dev. of interarrival times / a Results of chapter 7 or 8 do not apply, require simulation or more complicated models 15
Analyzing the Service Times Seasonality and Variability Call duration [minutes].5 Week-end averages 1.5 1 Week-day averages 0.5 0 0:00 3:00 Time of the day Frequency 800 CV p : Coefficient of variation of service times 600 00 00 0 Std. Dev = 11.6 Mean = 17. N = 061.00 Call durations [seconds] A 16
Computing the expected waiting time T q Inventory waiting I q Inventory in service I p Inflow Outflow Entry to system Begin Service Departure Waiting Time T q Service Time p Flow Time T=T q +p 17
Utilization FlowRate 1/ a Utilizatio n = u = = = Capacity 1/ p p a Example: Average Activity time=p=90 seconds Average Interarrival time=a=300 seconds Utilization=90/300=0.3=30% 18
The Waiting Time Formula Waiting Time Formula for Exponential Arrivals utilization CV + = a CV p Time in queue Activity Time 1 utilization Variability factor Utilization factor Service time factor Example: Average Activity time=p=90 seconds Average Interarrival time=a=300 seconds CV a =1 and CV p =1.333 Average time in queue 0.3 1 + 1.333 = 90 1 0.3 = 53.57 sec 19
Bank Teller Example An average of 10 customers per hour come to a bank teller who serves each customer in minutes on average. Assume exponentially distributed interarrival and service times. (a) What is the teller s utilization? (b) What is the average time spent in the queue waiting? (c) How many customers would be waiting for this teller or would be serviced by this teller on average? (d) On average, how many customers are served per hour? Answer: p= mins; a=6 mins=60/10 p a) u = = = 0.66 a 6 0.66 1 + 1 b) Tq = 8 1 0.66 = 1 1 c) I = * ( Tq + p) = * ( 8+ ) = a 6 d) If teller is busy wp/3, outputs15 per hour. If teller is idle wp1/3, outputs0 per hour. Averageoutput = (/3)*15+ (1/ 3)*0 = 10 per hour = Averageinput!!! 0
The Flow Time Increase Exponentially in Utilization Average flow time T Increasing Variability Theoretical Flow Time Utilization 100% 1
Computing T q with m Parallel Servers Inventory in the system I=I q + I p Inventory in service I p Inflow Inventory waiting I q Outflow Entry to system Begin Service Departure Waiting Time T q Service Time p Flow Time T=T q +p
Utilization with m servers Utilizatio n = u = FlowRate Capacity = 1/ a m / p = p a m 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% 3
Waiting Time Formula for Parallel Resources Approximate Waiting Time Formula for Multiple (m) Servers Time in queue Activity m ( m+ time utilization 1 utilization 1) 1 CVa + CV p Example: Average Activity time=p=90 seconds Average Interarrival time=a=11.39 seconds m=10 servers CV a =1 and CV p =1.333 90 (10+ 1) 1 0.79 Time in queue 10 1 0.79 T = T + p =.9 + 90 = 11.9 sec q 1 + 1.333 = 1.916 min =.9 sec
Online Retailer Example Customers send emails to a help desk of an online retailer every minutes, on average, and the standard deviation of the inter-arrival time is also minutes. The online retailer has three employees answering emails. It takes on average minutes to write a response email. The standard deviation of the service times is minutes. (a) (b) 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= mins; CV a =1; m=3; p= mins; CV p =0.5 a) Find T q. b) Find I q =(1/a)T q 5
Service Levels in Waiting Systems Fraction of customers who have to wait x seconds or less 1 0.8 Waiting times for those customers who do not get served immediately 90% of calls had to wait 5 seconds or less 0.6 0. Fraction of customers who get served without waiting at all 0. 0 0 50 100 150 00 Waiting time [seconds] Target Wait Time (TWT) Service Level = Probability{Waiting Time TWT}; needs distribution of waiting time Example: Deutsche Bundesbahn Call Center - now (003): 30% of calls answered within 0 seconds - target: 80% of calls answered within 0 seconds 6
Bank of America s Service Measures 7
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
Summary of the formulas 1. Collect the following data: - number of servers, m - activity time, p - interarrival time, a - coefficient of variation for interarrival (CV a ) and processing time (CV p ). Compute utilization: u= a p m 3. Compute expected waiting time ( Activity time T q utilizatio n = m 1 utilizatio m + 1) 1 n CV a + CV p. Based on T q, we can compute the remaining performance measures as Flow time T=T q +p Inventory in service I p =m*u Inventory in the queue=i q =T q /a Inventory in the system I=I p +I q 9
Staffing levels Cost of direct labor per serviced unit Totalwagesper time m x (wagesper time) Cost of Direct Labor= = = Flowrate= Arrivalrate 1/a p x (wagesper time) u 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 Utilization = u = Cost of Recall T Direct Labor = 1.916; p m x a = 90 10x11.39 = 0.79 1.5 min/call x (16.66 cents/min) = = 31.6 cents/call 0.79 Cost of line charge per call = 1.916x0.05 = $0.0958/ call 30
Cost of Staffing Levels m u Labor cost per call Line cost charge per call Total cost per call 8 1.358 9 0.01 10 0.790 0.316 0.0958 0.1 11 0.33 1 0.593 13 0.887 1 0.5193 15 0.5503 The optimal staffing level m=10 31
Staffing Levels under various Interarrival Times Number of customers Per 15 minutes Number of CSRs 160 10 10 100 80 60 0 17 16 15 1 13 1 11 10 9 87 6 5 0 3 1 0 Time 3 0:15 :00 3:5 5:30 7:15 9:00 10:5 1:30 1:15 16:00 17:5 19:30 1:15 3:00
The Power of Pooling Independent Resources x(m=1) Waiting Time T q 70.00 60.00 m=1 50.00 0.00 30.00 m= Pooled Resources (m=) 0.00 10.00 0.00 60% 65% Implications: + balanced utilization m=5 m=10 70% 75% 80% 85% 90% 95% Utilization u + Shorter waiting time (pooled safety capacity) - Change-overs / set-ups
Service-Time-Dependent Priority Rules 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? A 9 min. B C min. D Service times: A: 9 minutes B: 10 minutes C: minutes D: 8 minutes 19 min. C 1 min. A 3 min. D 1 min. B Total wait time: 9+19+3=51min Total wait time: +1+1=37 min Problem of having true processing times
Service-Time-Independent Priority Rules 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 W(t) p p 1 First arrival Second arrival Last departure Time t
Service-Time-Independent Order does not affect the waiting time Order: 1-1- Order: 1--1 W(t) W(t) First finishes Time t Second finishes Time t 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).
Summary Interarrival and Service Times and their variability Obtaining the average time spent in the queue Pooling of server capacities Priority rules 37