Managing Flow Variability: Safety Capacity OPSM301 Spring 2012 Class 19: Managing Flow Variability: Safety Capacity or Management of Queues

Economics of the Waiting Line Problem A central problem in many service settings is the management of waiting time Reducing waiting time costs money Waiting time also costs money When people waiting are customers, it is difficult to value their time Lost sales is one value

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

The Service Process Customer Inflow (Arrival) Rate (Ri) () Inter-arrival Time = 1 / Ri Processing Time Tp (unit load) Processing Rate per Server = 1/ Tp (µ) Number of Servers (c) Number of customers that can be processed simultaneously Total Processing Rate (Capacity) = Rp= c / Tp (cµ)

Operational Performance Measures processing waiting () Ri e.g 10 /hr R () 10 /hr Tw? 10 min, Rp=12/hr Flow time T = Tw + Tp (waiting+process) Inventory I = Iw + Ip Flow Rate R = Min (Ri, Rp) Stable Process = Ri < Rp,, so that R = Ri Little’s Law: I = R  T, Iw = R  Tw, Ip = R  Tp Capacity Utilization = Ri / Rp < 1 Safety Capacity = Rp – Ri Number of Busy Servers = Ip= c = Ri  Tp

Flow Times with Arrival Every 4 Secs (Service time=5 seconds) Customer Number Arrival Time Departure Time Time in Process 1 5 2 4 10 6 3 8 15 7 12 20 16 25 9 30 24 35 11 28 40 32 45 13 36 50 14 What is the queue size? Can we apply Little’s Law? What is the capacity utilization?

Flow Times with Arrival Every 6 Secs (Service time=5 seconds) Customer Number Arrival Time Departure Time Time in Process 1 5 2 6 11 3 12 17 4 18 23 24 29 30 35 7 36 41 8 42 47 9 48 53 10 54 59 What is the queue size? What is the capacity utilization?

Effect of Variability What is the queue size? Customer Number Arrival Time Processing Time Time in Process 1 7 2 10 3 20 4 22 5 32 8 6 33 14 36 15 43 16 9 52 12 54 11 Average=6 Average=5 What is the queue size? What is the capacity utilization?

Effect of Synchronization Customer Number Arrival Time Processing Time Time in Process 1 8 2 10 3 20 4 22 7 5 32 6 33 36 43 9 52 54 What is the queue size? What is the capacity utilization?

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 inter-arrival and processing times can be synchronized (correlated), queues and waiting times will be reduced

Summary: Causes of Delays and Queues High Unsynchronized Variability in Interarrival Times Processing Times High Capacity Utilization r = Ri / Rp, or Low Safety Capacity Rs = Rp – Ri, due to High Inflow Rate Ri Low Processing Rate Rp = c/ Tp (i.e. long service time, or few servers)

The Queue Length Formula Utilization effect Variability effect x where  Ri / Rp, where Rp = c / Tp, and CVi and CVp are the Coefficients of Variation (Standard Deviation/Mean) of the inter-arrival and processing times (assumed independent)

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

In words: In high utilization case: small decrease in utilization yields large improvement in response time This marginal improvement decreases as the slack in the system increases

Deriving Performance Measures from Queue Length Formula Use the formula to find Iw Tw = Iw /R T = Tw + Tp Ip = Tp R I =Iw + Ip

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)

Example :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)

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

Current Situation: waiting times Accounting, Operations, Finance: T =processing time+waiting time =0.25 hrs. + 0.25 hrs =0.5 hrs=30 min Marketing: T =processing time+waiting time =0.25 hrs. + 0.75 hrs =1 hr=60 min

Proposal: Secretarial Pool Accounting 2 Finance 2 3 Marketing 9 requests/hour 2 Operations Arrival rate=R=9/hr Tp=1/4 hr, Rp=c/Tp=16/hr Utilization=Ri/Rp=9/16

Proposed System: Secreterial pool T =processing time+waiting time =0.25 hrs. + 0.04 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

Effect of Pooling Pooled service capacity reduces waiting Ri/2 Server 1 Queue 1 Server 2 Queue 2 Ri/2 Ri Ri/2 Server 1 Pooled service capacity reduces waiting Ri Queue Server 2

Examples of pooling in business Consolidating back office work Call centers Single line versus separate queues

The impact of task integration (pooling) balances utilization... reduces resource interference... ...therefore reduces the impact of temporary bottlenecks there is more benefit from pooling in a high utilization and high variability process pooling is beneficial as long as it does not introduce excessive variability in a low variability system the benefits exceed the task time reductions due to specialization

Intuition building exercise Check the following website: Waiting Line Simulation (use internet explorer) http://archive.ite.journal.informs.org/Vol7No1/DobsonShumsky/security_simulation.php 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.

Exercise: Example 1 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. Ri=1/6 per min=10/hr Tp=4 min, c=1 Rp =15/hr =10/15=0.66 CVi=1, CVp=0 Exercise: 1. What if we have a human server, with CV=1? 2.What is the effect of buying a second machine?

Exercise Example 2: Computing Performance Measures Given Interarrival times: 10, 10, 2, 10, 1, 3, 7, 9, and 2 seconds Avg=6, stdev=3.937, Ri =1/6 Processing times: 7, 1, 7, 2, 8, 7, 4, 8, 5, 1 seconds Avg=5, stdev=2.8284 c = 1, Rp =1/5 Compute Capacity Utilization r = Ri / Rp = 5/6=0.833 CVi = 3.937/6 = 0.6562 CVp = 2.8284/5 = 0.5657 Queue Length Formula Iw = 1.5633 Hence Tw = Iw / R = 9.38 seconds, and Tp = 5 seconds, so T = 14.38 seconds, so I = RT = 14.38/6 = 2.3966 customers in the system

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, Ri =1/6 Processing times: 7, 1, 7, 2, 8, 7, 4, 8, 5, 1 Avg=5, stdev=2.8284 c = 2, Rp =2/5 Compute Capacity Utilization r = Ri / Rp = 0.4167 CVi = 3.937/6 = 0.6562 CVp = 2.8284/5 = 0.5657 Queue Length Formula Iw = 0.07536 Hence Tw = Iw / R = 0.45216 seconds, and Tp = 5 seconds, so T = 5.45216 seconds, so I = RT = 5.45216/6 = 0.9087 customers in the system

Exercise: Capacity planning A bank would like to improve its drive-in service by reducing waiting and transaction times. Average rate of customer arrivals is 30/hour. Customers form a single queue and are served by 4 windows in a FCFS manner. Each transaction is completed in 6 minutes on average. The bank is considering to lease a high speed information retrieval and communication equipment that would cost 30 TL per hour. The facility would reduce each teller’s transaction time to 4 minutes per customer. If our manager estimates customer cost of waiting in queue to be 20 TL per customer per hour, can she justify leasing this equipment? b. The competitor provides service in 8 minutes on average. If the bank wants to meet this standard, should it lease the new equipment?

Want to eliminate as much variability as possible from your processes: how? specialization in tasks can reduce task time variability standardization of offer can reduce job type variability automation of certain tasks IT support: templates, prompts, etc. Incentives Scheduled arrivals to reduce demand variability Initiatives to smoothen arrivals

Want to reduce resource interference in your processes: how? smaller lotsizes (smaller batches) better balanced line by speeding-up bottleneck (adding staff, changing procedure, different incentives, change technology) through cross-training eliminate steps buffers integrate work (pooling)