OPSM 301: Operations Management Session 13-14: Queue management Koç University Graduate School of Business MBA Program Zeynep Aksin

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

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

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

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

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)

Variability? Histogram of service times at a clinic Some patients take very long, some very short service time changes between 1 min - 20 min Variability is high In this example, arrival times have exponential distribution

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

Distributions  Exponential distribution: when arrivals at a service facility occur in a purely random fashion –The probability function is f(t) = λe -λt  Poisson distribution: where one is interested in the number of arrivals during some time period T –The probability function is

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

The Queue Length Formula Utilization effectVariability effect x where  R i / R p, where R p = c / T p, and CV i and CV p are the Coefficients of Variation (Standard Deviation/Mean) of the inter-arrival and processing times (assumed independent)

Variability Increases Average Time in System T Utilization (ρ)  100% TpTp Throughput- Delay Curve We must have slack capacity ρ < 1

Deriving Performance Measures from Queue Length Formula  Use the formula to find I w  T w = I w /R  T = T w + T p  I p = T p R  I=I w + I p

How can we reduce waiting?  Reduce utilization: –Increase capacity: faster servers, better process design, more servers  Reduce variability –Arrival: Appointment system –Service:Standardization of processes, automation  We can control arrivals –Short lines (express cashiers) –Specific hours for specific customers –Specials (happy hour)

Suggestions for Managing Queues  Segment the customer  Train your servers to be friendly  Inform your customers of what to expect  Try to divert the customer’s attention when waiting  Encourage customers to come during slack periods

Computer Simulation of Waiting Lines  Some waiting line problems are very complex  Have assumed waiting lines are independent  When a services becomes the input to the next, we can no longer use the simple formulas  Also true for any problem where conditions do not meet the requirements of the equations  Here, must use computer simulation

Example 1: 17 An automated pizza vending machine heats and dispenses a slice of pizza in exactly 4 minutes. Customers arrive at a rate of one every 6 minutes with the arrival rate exhibiting a Poisson distribution. Determine: A) The average number of customers in line. B) The average total waiting time in the system. R i =1/6 per min=10/hr T p =4 min, c=1 R p =15/hr  =10/15=0.66 CV i =1, CV p =0 Exercise: 1. What if we have a human server, with CV=1? 2.What is the effect of buying a second machine?

Example 2: Computing Performance Measures  Given –Interarrival times: 10, 10, 2, 10, 1, 3, 7, 9, and 2 seconds Avg=6, stdev=3.937, R i =1/6 –Processing times: 7, 1, 7, 2, 8, 7, 4, 8, 5, 1 seconds Avg=5, stdev= –c = 1, R p =1/5  Compute –Capacity Utilization  = R i / R p = 5/6=0.833 –CV i = 3.937/6 = –CV p = /5 =  Queue Length Formula –I w =  Hence –T w = I w / R = 9.38 seconds, and T p = 5 seconds, so –T = seconds, so –I = RT = 14.38/6 = customers

Example 2:Effect of Increasing Capacity  Assume an indentical server is added (c=2). Given –Interarrival times: 10, 10, 2, 10, 1, 3, 7, 9, and 2 Avg=6, stdev=3.937, R i =1/6 –Processing times: 7, 1, 7, 2, 8, 7, 4, 8, 5, 1 Avg=5, stdev= –c = 2, R p =2/5  Compute –Capacity Utilization  = R i / R p = –CV i = 3.937/6 = –CV p = /5 =  Queue Length Formula –I i =  Hence –T w = I w / R = seconds, and T p = 5 seconds, so –T = seconds, so –I = RT = /6 =

Example 3:Effect of pooling  4 Departments and 4 Departmental secretaries  Request rate for Operations, Accounting, and Finance is 2 requests/hour  Request rate for Marketing is 3 requests/hour  Secretaries can handle 4 requests per hour  Marketing department is complaining about the response time of the secretaries. They demand 30 min. response time  College is considering two options: –Hire a new secretary –Reorganize the secretarial support  Assume inter-arrival time for requests and service times have exponential distribution (i.e. CV=1)

21 Current Situation Accounting Finance Marketing Operations 2 requests/hour 3 requests/hour 2 requests/hour 4 requests/hour

22 Current Situation: waiting times T =processing time+waiting time =0.25 hrs hrs =0.5 hrs=30 min Accounting, Operations, Finance: Marketing: T =processing time+waiting time =0.25 hrs hrs =1 hr=60 min

23 Proposal: Secretarial Pool Accounting Finance Marketing Operations 9 requests/hour Arrival rate=R=9/hrTp=1/4 hr, R p =c/T p =16/hr Utilization=Ri/Rp=9/16

Proposed System: Secreterial pool T =processing time+waiting time =0.25 hrs hrs =0.29 hr=17.4 min In the proposed system, faculty members in all departments get their requests back in 17 minutes on the average. (Around 50% improvement for Acc, Fin, and Ops and 75% improvement for Marketing). Pooling improves waiting times by ensuring effective use of capacity

Server 1 Queue 1 Server 2 Queue 2 Server 1 Queue Server 2 Effect of Pooling RiRi RiRi R i /2 Pooled service capacity reduces waiting

Performance Improvement Levers  Capacity Utilization / Safety Capacity –Demand Management (arrival rate) Peak load pricing –Increase Capacity (processing rate) Number of Servers (scale) Processing Rate (speed)  Variability Reduction –Arrival times Scheduling, Reservations, Appointments –Processing times Standardization, Specialization, Training  Synchronization –Matching capacity with demand

Bonus Individual Assignment (0.5 points)  Check the following website:  Waiting Line Simulation (use internet explorer) Waiting Line Simulation   Run six different examples. Suggestion (you can use different numbers): –Arrival rate=9, service rate=10, CV=0, CV=1, CV=2 CV=0.5 –Arrival rate =9, service rate=12 CV=1 CV=0.5  write down the parameters and the average performance measures to observe the effect of utilization and variability on waiting times. Compare the simulation output with the results you find using formulas. Note the effect of variability and utilization.  Due on Tuesday after the exam 27