Queueing Theory Professor Stephen Lawrence Leeds School of Business University of Colorado Boulder, CO

2 Outline General Queues General distributions Multiple servers G/G/k queues M/M/k queues G/M/1 queues D/D/k queues Other queueing systems Simple Queues Intro to queues Queue assumptions Exponential distribution M/M/1 queues Service/cost tradeoffs Managerial implications

3 Service Queuing Analysis Arrival Rate (  Average Number in Queue ( L q ) Avg Time in System ( W ) Avg Number in System ( L ) Average Wait in Queue ( W q ) Rate (  Departure

4 Principal Queue Parameters Arrival Process Service Process Number of Servers Queue Discipline

5 1. Arrival Process In what pattern do jobs / customers arrive to the queueing system? Distribution of arrival times? Batch arrivals? Finite population? Finite queue length? Poisson arrival process often assumed Many real-world arrival processes can be modeled using a Poisson process

6 2. Service Process How long does it take to service a job or customer? Distribution of arrival times? Rework or repair? Service center (machine) breakdown? Exponential service times often assumed Works well for maintenance or unscheduled service situations

7 3. Number of Servers How many servers are available? Single Server Queue Multiple Server Queue

8 4. Queue Discipline How are jobs / customers selected from the queue for service? First Come First Served (FCFS) Shortest Processing Time (SPT) Earliest Due Date (EDD) Priority (jobs are in different priority classes) FCFS default assumption for most models

9 Queue Nomenclature X / Y / k (Kendall notation) X = distribution of arrivals (iid) Y = distribution of service time (iid) M = exponential (memoryless) E m = Erlang (parameter m) G = general D = deterministic k = number of servers

10 The Poisson Distribution The interarrival times the population of a Poisson process are exponentially distributed…

11 Exponential Distribution Simplest distribution Single parameter (mean) Standard deviation fixed and equal to mean Lacks memory Remaining time exponentially distributed regardless of how much time has already passed Interarrival times of a Poisson process are exponential

12 Exponential Distribution Exponential Density Mean = m Std Dev = s =m f(t) 1/ m t m

M/M/1 Queues

14 M/M/1 Assumptions Arrival rate of  Poisson distribution Service rate of  Exponential distribution Single server First-come-first-served (FCFS) Unlimited queue lengths allowed “Infinite” number of customers

15 M/M/1 Operating Characteristics Utilization (fraction of time server is busy) Average waiting times Average number waiting

16 Example Boulder Reservoir has one launching ramp for small boats. On summer weekends, boats arrive for launching at a mean rate of 6 boats per hour. It takes an average of s=6 minutes to launch a boat. Boats are launched FCFS.  = 6 /hr  = 6/10 = 0.6 or 60% L =  = 6/(10-6) = 1.5 boats L q = L  = 1.5(0.6) = 0.9 boats W = 1/  = 1/(10-6) = 0.25 hrs or 15 mins W q = W  = 0.25(0.6) = 0.15 hrs or 9 mins  = 1/s =1/6 = 0.167/min or 10 /hr

17 Example (cont.) During the busy Fourth of July weekend, boats are expected to arrive at an average rate of 9 per hour.  = 9 /hr  = 1/s =1/6 = 0.167/min or 10 /hr  = 9/10 = 0.9 or 90% L =  = 9/(10-9) = 9 boats L q = L  = 9(0.6) = 5.4 boats W = 1/  = 1/(10-9) = 1.0 hrs or 60 mins W q = W  = 1(0.9) = 0.9 hrs or 54 mins

18 Another Example The personal services officer of the Aspen Investors Bank interviews all potential customers to ascertain that they have sufficient net worth to become clients. Potential customers arrive at a rate of nine every 2 hours according to a Poisson distribution, and the officer spends an average of twelve minutes with each customer reviewing their portfolio with an exponential distribution. Determine the principal operating characteristics for this system.

19 Queue Simulation Averages are deceptive Simulation of M/M/1 queue shows the effect of variance Excel spreadsheet queue simulation Available on course website

20 Managerial Implications Low utilization levels provide better service levels greater flexibility lower waiting costs (e.g., lost business) High utilization levels provide better equipment and employee utilization fewer idle periods lower production/service costs Must trade off benefits of high utilization levels with benefits of flexibility and service

21 Flexibility/Utilization Trade-off Utilization   = 1.0  = 0.0 L L q W W q High utilization Low ops costs Low flexibility Poor service Low utilization High ops costs High flexibility Good service

22 Cost Trade-offs   = 0.0 Cost Combined Costs Cost of Waiting Cost of Service Sweet Spot – Min Combined Costs 

G/G/k Queues

24 G/G/k Assumptions General interarrival time distribution with mean a=1/ and std. dev. = s a General service time distribution with mean p=1/  and std. dev. = s p Multiple servers (k) First-come-first-served (FCFS) “Infinite” calling population Unlimited queue lengths allowed

25 General Distributions Two parameters Mean (m) Std. dev. (s  ) Examples Normal Weibull LogNormal Gamma f(t) t m s Coefficient of Variation cv = s/m

26 G/G/k Operating Characteristics Average waiting times (approximate) Average number in queue and in system

27 Alternative G/G/k Formulation Since 1/  = p Variance Term Congestion Term Service Time Term

28 G/G/k Analyzed Suppose m =s (cv = 1) and k =1 (M/M/1) M/M/1 result! for both arrival & service processes

29 G/G/k Analyzed Waiting time increase with square of arrival or service time variation Decrease as the inverse of the number of servers

30 G/G/k Variance Analyzed Waiting times increase with the square of the coefficient of variance No variance, no wait! WqWq c

31 M/M/2 Example The Boulder Parks staff is concerned about congestion during the busy Fourth of July weekend when boats are expected to arrive at an average rate of 9 per hour and take 6 minutes per boat to unload. Boulder is considering constructing a second temporary ramp next to the first to relieve congestion. What will be its effect?

32 Another G/G/k Example Aspen Investors Bank wants to provide better service to its clients and is considering two alternatives: 1.Add a second personal services officer 2.Install a computer system that will quickly provide client information and reduce service time variance (service time standard deviation cut in half). Recall that customers arrive at a rate of 4 per hour and are serviced at a rate of 5 per hour.

Other Queueing Models

34 Other Queueing Behavior Server Queue (waiting line) Customer Arrivals Customer Departures Customer balks (never enters queue) Customer reneges (abandons queue) Wait too long? Line too long?

35 Disney Queues

36 Waiting Line Psychology 1.Waits with unoccupied time seem longer 2.Pre-process waits are longer than process 3.Anxiety makes waits seem longer 4.Uncertainty makes waits seem longer 5.Unexplained waits seem longer 6.Unfair waits seem longer than fair waits 7.Valuable service waits seem shorter 8.Solo waits seem longer than group waits Maister, The Psychology of Waiting Lines, teaching note, HBS

37 Queues and Simulation Only simple queues can be mathematically analyzed “Real world” queues are often very complex multiple servers, multiple queues balking, reneging, queue jumping machine breakdowns networks of queues,... Need to analyze, complex or not Computer simulation !