OMG Operations Management Spring 1997 CLASS 4: THE IMPACT OF VARIABILITY Harry Groenevelt

2 Agenda The Puff Line Sources of Variability Introduction to Queues The Physics of Queues –Impact of high utilization –Economies of scale –Quantifying the impact of variability Summary of Insights

3 Sources of Variability In any process there is variability in: l demand from external or internal customers l processing time within the system The impact of variability can be especially severe in service systems, which cannot build inventory to prepare for the ‘peak’

4 Sources of Demand Variability: Examples IRS service center: seasonal variability Mutual fund service center: early evening peak Variability created by other processes within the firm: –production batches (Shot-Peening,...) –transfer batches (e.g., filling a cart, truck or boat) –surges at airline ‘hubs’ –others….

5 Servers (s) system queue departures arrivals ( customers/hour)  customers/hr./server Introduction to Queues: Notation arrival rate ( ) in customers/hour service rate (  ) in customers/hour (avg. time for one customer = ____) # servers = s; Therefore, capacity = ____

6 Introduction to Queues: Examples servicecustomerqueue‘service’ facilityarrives..locationprocess health clinicfrontwaitingtreatment deskroom on-call computer consultant AOL ‘modem farm’

7 average number in queue average number in system wait in queue system queue wait in system prob(waiting time > t) Introduction to Queues: Performance Measures

8 Performance Measures for Queues Throughput = rate of customers served Utilization (  ) = throughput / capacity = proportion of all server time spent working = avg. number being served / number of servers Load Factor = arrival rate / capacity For the system on page 5, Throughput = ______ utilization = load factor = _____ But … when is throughput  arrival rate, utilization  load factor?

9 ‘memoryless’ arrivals ‘memoryless’ servers one server Introduction to Queues: Specifying Variability Must specify variability of arrivals and service times. One example: the M/M/1 queue –Time between arrivals exponentially distributed (‘Poisson’ arrival process) –Service times exponentially distributed –All customers served in order of arrival –Arrival and service rates constant (stationary system)

10 Avg. time between arrivals = 1/ = 0.2 hours = 12 minutes Avg. service time = 1/  = 10 minutes Physics of Queues: Example ‘Rapid Oil Change’: –one service bay –Poisson arrivals with rate 5 cars/hour ( ) –Exponential service times, mean = 10 min. (1/  )

11 utilization  0 1 avg. number in system Physics of Queues: Impact of High Utilization For an M/M/1 system average time in system = 1/(  – ) = 1/(6 cars/hour – 5 cars/hour) = 1 hour load factor = /  =  = 5/6 average number in system = /(  – ) =  /(1–  ) = (5/6) / (1–5/6) = 5 cars.

12 Physics of Queues: Utilization As utilization approaches 1, average time in system, wait in queue, number in system and number in queue all rise dramatically. This effect seen in any system, including: –M/M/s (multiple servers with ‘snake’ line) –G/G/s (multiple servers with ‘snake’ line, General arrival or service distribution)

13 Physics of Queues: Economies of Scale M/M/s Example: Rapid Oil Change Same as in first example except: –2 service bays (twice as many) –Poisson arrivals with rate 10 per hour (twice as high) What is this system’s utilization? Equations for M/M/s are not as simple, so we implement them in Excel...

14 M/M/s example results (from QMACROS): single bay two bays Physics of Queues Note the difference between 1 and 2 bay systems!

Number of Servers Pr{Wait > 5 minutes} Pr{Wait > 5 minutes} (see left scale) (Avg) Wait in Queue (hrs) (Avg) Wait in Queue (hrs) (see right scale) Physics of Queues M/M/s example: Economies of Scale –Vary number of servers and raise arrival rate proportionally (load factor always 5/6 = )

16 The Physics of Queues: Impact of Variability G/G/s model (same as M/M/s model, except): –General service time distribution –General inter-arrival time distribution

17 Impact of variability G/G/s model –For arrival process specify: Arrival Rate Coefficient of Variation of inter-arrival time distribution (cv(A)) –For service time distribution specify: Service Rate Coefficient of Variation of service time distribution (cv(S))

18 Impact of variability Reminder: if X is a random variable with mean  and standard deviation , then its Coefficient of Variation = cv(X) =  /  For exponential random variables: –Coefficient of Variation = 1 For deterministic random variables: –Coefficient of Variation = ________

19 An approximation good for ‘congested’ systems: What happens as arrivals become more ‘lumpy’? As service times become more variable? Similar effect for G/G/s (try in QMACROS) Impact of variability: G/G/1

20 Summary of Insights High utilization causes congestion, high WIP and long lead times Variability causes congestion, high WIP and long lead times At the same utilization, a larger system will perform better than a smaller system - or - smaller systems must have lower utilization to perform as well as larger systems