2019/5/12 Shin‐Ming Guo Managing Waiting Lines NKFUST • The Economies of Waiting • Features of Queuing Systems • Estimating Waiting Times • Waiting Line Management What are waiting lines and why do they form? Answer: Waiting Lines form due to a temporary imbalance between the demand for service and the capacity of the system to provide the service. Service system Customer Served population customers Waiting line Service facilities Priority rule 1
2019/5/12 The Economies of Waiting Total Cost per hour = Cost of Capacity per hour + Cost of Customer Waiting Cost Cost of Capacity Total Cost minimum Cost of Waiting Small Service Capacity Large 3 I. The Operation of a Typical Call Center Call center Answered Incoming Calls Sales reps calls Calls processing calls on Hold At peak, 80% of calls dialed received a busy signal. Customers getting through had to wait on average 10 min. . Blocked calls Abandoned calls (busy signal) (tired of waiting) Holding cost Lost throughput Cost of capacity $$$ Revenue $$$ Lost goodwill Cost per customer Lost throughput (abandoned) 4 2
2019/5/12 A “Perfect or Somewhat Odd” Call Center Arrival Service Caller Time Time 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 5 A More Realistic Service Process caller 1 caller 3 caller 5 caller 7 caller 9 caller 11 Caller Arrival Service Time Time caller 2 caller 4 caller 6 caller 8 caller 10 caller 12 1 0 5 Time 2 7 6 3 9 7 7:00 7:10 7:20 7:30 7:40 7:50 8:00 4 12 6 5 18 5 3 6 22 2 Number of cases 7 25 4 2 8 30 3 9 36 4 1 10 45 2 11 51 2 0 2 min. 3 min. 4 min. 5 min. 6 min. 7 min. 12 55 3 Service times 6 3
2019/5/12 Variability Leads to Waiting Time Arrival Service Caller Time Time Service time 1 0 5 2 7 6 3 9 7 4 12 6 5 18 5 Wait time 6 22 2 7 25 4 8 30 3 9 36 4 7:00 7:10 7:20 7:30 7:40 7:50 8:00 10 45 2 5 11 51 2 4 12 55 3 3 Inventory (callers on line) 2 1 0 7:00 7:10 7:20 7:30 7:40 7:50 8:00 7 Variability: Where does it come from? Tasks • Inherent variation • Lack of SOPs • Quality (scrap / rework) Processing Buffer Input • Random arrivals (randomness is the rule, Resources not the exception) • Breakdowns / Maintenance • Unpredicted Volume swings • Operator absence • Product Mix • Set‐up times • Incoming quality 8 4
2019/5/12 II. Essential Features of Queuing Systems Renege Arrival Queue Departure process discipline Service Calling Queue process population configuration Balk No future need for service 9 Customer Arrival Process Arrival process Static Dynamic Random arrival Customer‐ Random Facility‐ exercised arrivals with rate varying controlled control constant rate with time Accept/Reject Price Appointments Reneging Balking 5
2019/5/12 Analyzing Inter-Arrival Times Arrival Call Interarrival Time Time, AT i Call 1 Call 2 Call 3 Call 4 Call 5 Call 6 Call 7 IA i =AT i+1 -AT i 1 6:00:29 00:23 2 6:00:52 01:24 3 6:02:16 00:34 4 6:02:50 6:00 6:01 6:02 6:03 6:04 6:05 6:06 Time 02:24 5 6:05:14 00:36 6 6:05:50 IA 1 IA 2 IA 3 IA 4 IA 5 IA 6 00:38 7 6:06:28 standard deviation CV Coefficien t of Variation a mean 11 Modeling Customer Arrivals Random (Poisson) arrivals Customers arriving independently from each other follow exponential inter‐arrival times. ( T ) n P n = CV 1 for n = 0, 1, 2,… a n ! P n =Probability of n arrivals in T time periods = Average numbers of arrivals per period 6
2019/5/12 Temporal Variation in Arrival Rates An arrival process is not stationary if the average number of arrivals in any given time interval is not fixed over the entire time period. Queue Configuration 7
2019/5/12 Multiple Queues vs. Single Queue Multiple Queues ? Single queue Take a Number 3 4 2 8 6 10 12 7 11 9 5 Enter 15 Queue Discipline Queue discipline Static Dynamic (FCFS rule) selection Selection based based on status on individual of queue customer attributes Number of Processing time customers Round robin Priority Preemptive of customers waiting (SPT rule) 8
2019/5/12 Service Process and Customer Involvement 17 Histograms of Service Times 1/ μ = mean service time μ = service rate CV s 1 18 9
2019/5/12 III. Estimating Wait Times waiting line 1. Mean arrival rate: arrival departure 2. Number of parallel servers: m Begin 3. Mean service rate: Entry to system End Service Service 4. Utilization: m W long time averages 5. Mean time in queue: q 1 6. Mean time in system: W W s q 7. Mean number of customers in queue: L W q q 8. Mean number of customers in the system: L W L m s s q Reducing Variability to Reduce Waiting 2 2 1 CV CV 2 2 2 W a s L q 1 2 q 2 ( 1 ) Service time factor Utilization factor Variability factor Reduce Arrival Variability • appointment/reservation: how to handle late arrivals or no‐shows • encourage customers to avoid peak hours. Reduce Service Time Variability • training and technology • limit service selection • reduce customer involvement 20 10
2019/5/12 100% Service Utilization? Single server 100 With: L Then: s 1 10 L s 8 0 0 6 0.2 0.25 0.5 1 4 0.8 4 2 0.9 9 0 1.0 0.99 99 0 Multiple, Parallel Resources with One Queue number in service number waiting L q departure arrival Entry to system Begin Service Departure Time in queue W q Service Time 1/ μ Time in system W s =W q + 1/μ 22 11
2019/5/12 Waiting Time for Multiple, Parallel Resources Under the assumption that arrival rate 1 Utilizatio n m service rate we approximate the average waiting time as 2 ( m 1 ) 1 2 2 utilizatio n CV CV 1/ W a s q m 1 utilizatio n 2 Service time factor Utilization factor Variability factor 23 Staffing over the Course of a Day Number of customers Number of Per 15 minutes CSRs 160 160 17 16 140 140 15 14 120 120 13 12 100 100 11 10 80 80 9 8 60 60 7 6 40 40 5 4 20 20 3 2 0 0 1 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 10:45 10:45 12:30 12:30 14:15 14:15 16:00 16:00 17:45 17:45 19:30 19:30 21:15 21:15 23:00 23:00 24 12
2019/5/12 IV. Waiting Times & Customer Satisfaction Reducing average waiting time does not guarantee customer satisfaction. A small percentage of customers may experience long waits and complain bitterly. Solution: service guarantee and/or service recovery 25 Perception of Waiting and Service Level Satisfaction Perceived Wait Time 1 Amount of time customers 0.8 believe they have waited sensitivity prior to receiving service. 0.6 Has a greater effect on 0.4 customer satisfaction than 0.2 actual waiting time 0 t* perceived wait threshold Service Level = Probability{Perceived Wait Time Target Wait Time } 26 13
2019/5/12 Psychology of Waiting • That Old Empty Feeling: Unoccupied time goes slowly • A Foot in the Door: Pre‐service waits seem longer that in‐service waits • The Light at the End of the Tunnel: Reduce anxiety with attention • Excuse Me, But I Was First: Social justice with FCFS queue discipline • They Also Serve, Who Sit and Wait: Avoids idle service capacity Factors Affecting Perceived Wait Times Server‐Related Factors Customer‐Related Factors Passive vs. active waits Solo versus group waits Unfair vs. fair waits Waits for more valuable versus less valuable Uncomfortable vs. services comfortable waits Customer’s own tolerance Unexplained vs. explained waits Unproductive vs. Productive waits 28 14
2019/5/12 Fair vs. Unfair Waits Suggestions for Managing Queues 1. Determine an acceptable waiting time for your customers 2. Inform your customers of what to expect 3. Try to divert your customer’s attention 4. Segment customers (Din Tai Fung, pay more) 5. Encourage customers to use slack periods 6. Appointment (no show?) 7. Provide limited service or self‐service 8. Provide flexible service hours or internet access 9. Technology: bar‐code, OCR, RFID (ETC) , beeper, fast Lane, customer database 10. Take a long‐term perspective (redesign the system) 30 15
2019/5/12 Segment Customers 31 Summary Variability is the norm, not the exception ‐ understand where it comes from and eliminate what you can Variability leads to waiting times although utilization<100% Operations benefit from flexibility in capacity Demand can exhibit seasonality → Time varying capacity Pooling resources can reduce waiting times Managing customers’ perceived wait times 32 16
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