1. Lines and Waiting Waiting and Service Quality A Quick Look at Queuing Theory Utilization versus Variability Tradeoff Managing Perceptions... “Every day I get in the queue, to get on the bus that takes me to you...” Pete Townsend, Magic Bus

2. Cultural Attitudes “Americans hate to wait. So business is trying a trick or two to make lines shorter...” The New York Times September 25, 1988 “An Englishman, even when he is by himself, will form an orderly queue of one...” George Mikes “How to be an Alien” In the Soviet Union, waiting lines were used as a rationing device... What are your experiences with Waiting Lines?

3. The Big Picture m We know that inventory and waiting times are important. If inventory or line lengths go down, ROI goes up. m We can reduce these thru more capacity, but... If capacity goes up, ROI goes down m Thus, balancing them is critical. Only our current tools aren’t good enough to do this well. m Queuing theory can help us.

4. Waiting – Key to Service Quality Recall the definition of flow time: What causes waiting? Insufficient Capacity Mismatch of Capacity and Demand “Lumpiness” in arrivals waiting – a critical component of the customer’s perception of service! Time People/Hr Capacity Avg. Arrivals

5. Why are We Always Waaaiting? Assume that every hour a bus arrives with 30 passengers at a movie theater. The 1 ticket vendor can sell 1 ticket per minute. What is the vendor’s capacity? What is the average thruput of customers? What is the utilization? What is the average waiting time? What is the average line length? Even with less than 100% utilization, we get waiting from lumpiness in the arrival rate. Vendor Q

6. Queuing Theory Our analysis up till now has assumed smooth and deterministic flows. Thus can’t predict line length and waiting. However, queuing theory captures the probabilistic nature of arrival rates and by more accurately predicting waiting time allows us to: Estimate avg. waiting times and line lengths in steady state (If Capacity > Arrival Rate) Adjust capacity to meet service requirements But first we need to make a couple of assumptions: Arrival Rates: Poisson Process Service Times: Exponentially Distributed Maximum Line Lengths: Unlimited Empirical studies have shown that these assumptions are good approximations in many real-world situations.

7. Go Poisson! m Deterministic & smooth assumptions don’t often work. m But, bus example is too “lumpy” for most real- world situations. m Let’s assume instead that the customers arrive completely independently of each other. So knowing one customer’s arrival time tells us nothing about anyone else’s. m This results in a Poisson (French for Fish) process. Under this process, the time between arrivals is exponentially distributed. So called because the equivalent of the Bell Curve for this is exponentially declining. Time between Arrivals Probability Density Function Normal Distribtion (Bell Curve) Exp. Distribution

8. Queuing Terms queue: Line feeding a number of servers. server: Task or operation fed by a queue. arrival rate ( ): Mean number of arrivals per unit time (usually per hour or day). interarrival time: Time between arrivals. server rate (  ): Mean number of arrivals serviced by each individual server per unit time at 100% utilization. channels (M): Number of servers connected to an individual queue. congestion (  ): Measure of how “busy” the system is.  = M * utilization. phase: Individual queue plus its attached servers. There can be more than one of these in a network. coefficient of variation (Cv): Measure of “lumpiness” = Std.Dev./ Mean. The higher the Cv of interarrival times, the lumpier the process. An exponential distribution has a Cv of 100%. Bell curves typically have a CV < 33%.

9. Queuing Example Suppose, on the average, 48 customers arrive randomly each hour at a bank and it takes each teller an average of 6 minutes to process a customer. What is the average customer flow time? How many tellers should the bank have? One line or multiple lines? Analyzed as a deterministic system: Number of Tellers: Utilization: Avg. Waiting Time:  

10. Queuing Example (cont’d) However, most real systems have variability (are stochastic) and deterministic models can be quite misleading... Single-phase, multi-channel model:   Utilization:  = Number of Channels: Avg. Line Length (Lq): Avg. Wait Time (Wq = Lq /  watch the time units):

11. Utilization:  = Number of Channels: Avg. Line Length (Lq): Avg. Wait Time (Wq): Queuing Example (cont’d) Single-phase, single-channel model: (Assume that lines are entered at random) 

12. Queuing Example (cont’d) 6 teller single-phase, multi-channel model: 7 teller single-phase, multi-channel model: Utilization:  = Number of Channels: Avg. Line Length (Lq): Avg. Wait Time (Wq): Utilization:  = Number of Channels: Avg. Line Length (Lq): Avg. Wait Time (Wq):

13. Tradeoffs The waiting time in a process “blows up” when managers demand a high utilization... Rules of Thumb: utilization of 85-90% is a good average target “lumpier” arrivals result in longer waiting times More variable service times result in longer waits

14. 3 Helpful Laws  The merger of two Poisson arrival processes with mean arrival rates of 1 and 2 is also Poisson and has mean arrival rate equal to 1 + 2. m Also under our assumptions: Jackson’s law: The exit rate from a queue is also Poisson and has the same mean as the arrival rate. m And Little’s Law still applies as long as utilization < 100%.

15. Queuing in a Family Medical Practice Note: Although it is better in practice to linearly interpolate Lq, in the interests of time, we will instead pick the Lq associated with the closest value of  listed in the chart to the given queue’s.

16. Family Medical Practice Problems m What are the key assumptions you need to make to analyze the system? m Estimate (A) the average waiting time for each queue. (B) the average time in the system for each of the three patient paths. (C) the overall average time in the system (D) the avg. idle time per hour for each employee m What is the effect of the patient prioritization (First Come-First Serve vs. something else)?

17. Balancing Costs Service Quality is about balancing costs… Trade-off between: service quality (i.e. waiting time): which will eventually reduce revenue (increasing ROI) capacity: which will increase both variable and fixed costs (lowering ROI). By picking the appropriate design you can, in theory, find a “maximum” return on your service quality investment. However, there may be another way to “skin the cat”...

18. Beating the Tradeoff There are two sides to customer satisfaction  can try to manage perception... preprocess time feels longer than in-process time unoccupied time feels longer than occupied time uncertain waits are worse than certain waits unexplained waits are worse than explained waits anxiety makes the wait feel even longer unfair waits are worse than fair waits valuable service  waiting is more acceptable You aren’t restricted to the Cost-Quality Tradeoff game...Change the rules!

19. The Message An average process capacity greater than the average arrival rate does not ensure small lines and short waits! However if you can’t afford to change the actual wait time, you can make your customers think its shorter with the appropriate perception management techniques.