1. OPSM 301: Operations Management Session 19: Flow variability Koç University Zeynep Aksin zaksin@ku.edu.tr

2. Announcements  Midterm 2-December 14 at 18:30 CAS Z48, CAS Z08 –Does not include Midterm 1 topics –LP, Inventory, Variability (Congestion+Quality) –LP: from course pack –Inventory Ch6 excluding 6.7, Ch 7.1, 7.2, 7.3 –Chapter 8 excluding 8.6 and 8.8 (this week) –Chapter 9 (next week)

3. Components of the Queuing System Visually Customers come in Customers are served Customers leave

4. Flow Times with Arrival Every 4 Secs (Service time=5 seconds) Customer Number Arrival Time Departure Time Time in Process 1055 24106 38157 412208 516259 6203010 7243511 8284012 9324513 10365014 What is the queue size? Can we apply Little’s Law? What is the capacity utilization?

5. Customer Number Arrival Time Departure Time Time in Process 1055 26115 312175 418235 524295 630355 736415 842475 948535 1054595 Flow Times with Arrival Every 6 Secs (Service time=5 seconds) What is the queue size? What is the capacity utilization?

6. Customer Number Arrival Time Processing Time Time in Process 1077 21011 32077 42227 53288 633714 736415 843816 952512 1054111 Effect of Variability What is the queue size? What is the capacity utilization?

7. Customer Number Arrival Time Processing Time Time in Process 1088 21088 32022 42277 53211 63311 73677 84377 95244 105457 Effect of Synchronization What is the queue size? What is the capacity utilization?

8. Conclusion  If inter-arrival and processing times are constant, queues will build up if and only if the arrival rate is greater than the processing rate  If there is (unsynchronized) variability in inter-arrival and/or processing times, queues will build up even if the average arrival rate is less than the average processing rate  If variability in interarrival and processing times can be synchronized (correlated), queues and waiting times will be reduced

9. To address the “how much does variability hurt” question: Consider service processes  This could be a call center or a restaurant or a ticket counter  Customers or customer jobs arrive to the process; their arrival times are not known in advance  Customers are processed. Processing rates have some variability.  The combined variability results in queues and waiting.  We need to build some safety capacity in order to reduce waiting due to variability

10. Why is there waiting?  the perpetual queue: insufficient capacity-add capacity  the predictable queue: peaks and rush-hours- synchronize/schedule if possible  the stochastic queue: whenever customers come faster than they are served-reduce variability

11. A measure of variability  Needs to be unitless  Only variance is not enough  Use the coefficient of variation  C or CV=  / 

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

13. Specifications of a Service Provider Service Provider Leaving Customers Waiting Customers Demand Pattern Resources Human resources Information system other... Arriving Customers Satisfaction Measures Reneges or abandonments Waiting Pattern Served Customers Service Time

14. Distribution of Arrivals  Arrival rate: the number of units arriving per period –Constant arrival distribution: periodic, with exactly the same time between successive arrivals –Variable (random) arrival distributions: arrival probabilities described statistically Exponential distribution for interarrivals Poisson distribution for number arriving CV=1

15. Service Time Distribution  Constant –Service is provided by automation  Variable –Service provided by humans –Can be described using exponential distribution CV=1 or other statistical distributions

16. The Service Process  Customer Inflow (Arrival) Rate (R i ) ( ) –Inter-arrival Time = 1 / R i  Processing Time T p (unit load) –Processing Rate per Server = 1/ T p (µ)  Number of Servers (c) –Number of customers that can be processed simultaneously  Total Processing Rate (Capacity) = R p = c / T p (cµ)

17. Operational Performance Measures Flow time T=T w +T p (waiting+process) Inventory I= I w + I p Flow Rate R =Min (R i, R p  Stable Process= R i < R p,, so that R = R i Little’s Law: I = R  T, I w = R  T w, I p = R  T p Capacity Utilization  = R i / R p < 1 Safety Capacity = R p – R i Number of Busy Servers = I p = c  = R i  T p waiting processing ( ) R i e.g10 /hr R ( ) 10 /hr 10 min, R p =12/hr Tw?Tw?

18. Summary: Causes of Delays and Queues  High Unsynchronized Variability in –Interarrival Times –Processing Times  High Capacity Utilization  = R i / R p, or Low Safety Capacity R s = R p – R i, due to –High Inflow Rate R i –Low Processing Rate R p = c/ T p (i.e. long service time, or few servers)

19. The psychology of waiting  waiting as psychological punishment  keep the customer busy  keep them entertained  keep them informed  break the wait up into stages  in multi-stages, its the end that matters

20. The psychology of waiting  waiting as a ritual insult  sensitivity training  make initial contact  waiting as a social interaction  prevent injustice  improve surroundings  design to minimize crowding  get rid of the line whenever possible

21. Reducing perceived wait  Understand psychological thresholds  Distract customers (mirrors, music, information)  Get customers out of line (numbers, call-back)  Inform customers of wait (over-estimate)  Keep idle servers out of sight  Maintain fairness (FCFS)  Keep customers comfortable

22. Is a queue always bad?  queues as a signal for quality  doctors  business schools  restaurants  other people demand similar things  the advantage of being in

23. A solution: Add capacity to remove a persistent line?  You want others to be there to signal quality  Risks of being in versus out: its an unstable proposition!  Don’t want to relate everything to price

24. The challenge: matching demand and supply  changing number of servers  changing queue configuration  changing demand  managing perceptions