WAITING LINE MODELS Introduction
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1 WAITING LINE MODELS Introduction Professor Robert Saltzman Operations Analysis Queuing Analysis: The Study of Waiting Lines Lines are everywhere in the service sector: Grocery store checkout Airport bag check Call Center Lines also exist in the manufacturing sector: Work-in-process in a factory Orders waiting in a job shop Broken machines needing repair 1
2 Why Do Lines Form? On average, not enough service available Variability in customer behavior: They don t arrive at evenly-spaced points in time Their service needs may fluctuate a lot Variability in server behavior: Servers may work at an inconsistent rate Both sources of variability lead to lines A small case: Esprit Outlet Store Occasional sales for 2-3 days High customer response: $100K+ sales/day Even with 12 registers: Average wait sometimes exceeded an hour Number of people entering store must be regulated Causes? * Slow (incompetent?) clerks * Registers break down occasionally * Crisis atmosphere panic! 2
3 Esprit Outlet Store - continued What impact does this have? Unhappy, tired customers Lost current sales: people give up & leave Lost future sales: loss of good will What can be done? Managerial Issues: How long a wait is reasonable? How might more service improve the wait? What s the cost of better service? Elementary Queuing Systems 1. Population Source: Finite or * 2. Customers Arrive: Poisson process*, Deterministic, etc. 3. Queue area: How many lines? One* or many? Do they have finite or * capacity? 4. Selection from Queue: FCFS*, LIFO or some other priority rule? Is preemption allowed? 5. Service Facility: How many independent servers? One or many? How many phases of service? One* or many? What s the service time distribution? 3
4 Standard Notation a shorthand Many different-looking systems are similar in structure Label format: A/B/s A: Arrival distribution: {M for Poisson, } B: Service time distribution: {D, M, } s: Number of servers working in parallel: {1, 2, 3, } M/M/1 = Model A in our text M/M/s = Model B M/D/1 = Model C Performance measures formulas exist for many systems A Few Examples of Elementary Queuing Systems Model Examples A. M/M/1 Single copy machine in the library B. M/M/s Set of s = 3 copiers in the library C. M/D/1 A coffee vending machine Other variations (labels) exist & have been analyzed, e.g., M/M/s/K = finite capacity (K) multi-server system 4
5 Steady State Performance Measures After the system has been operating a while, it reaches steady state, i.e., the system s initial conditions have worn off Can collect data on it s long-run behavior Key Performance Measures of Average Behavior: W q : Average waiting time in the queue W s : Average waiting time in the system (Q + service) L q : Average # of customers in the queue L s : Average # of customers in the system (Q + service) M = Poisson Arrival Pattern Customers arrive randomly, but at an overall average rate of λ ( lambda ) per unit time, EX: One call every 3 min. => λ = 20 calls/hour Poisson is good for representing: Arrivals to a bank or store Phone calls made to 911 Cars passing an intersection Poisson is not good for describing: Students arriving to a particular class Items entering an assembly line Buses reaching a specific bus stop 5
6 M = Exponential Service Time Time required to serve a customer often follows an exponentially decreasing pattern, whose mean is 1/µ ( mu ) EX: 1/µ = 10 minutes/customer, so µ = customers/hour Gather service time data & make histogram: EXPO often fits service time of: Bank Tellers Checkout Clerks Telephone Talkers EXPO probably poor for: Men s haircuts Car wash Time (min.) Machine-based service 2 Key Steady-State Relationships On average: 1) Wait in System = Wait in Line + Service Time: W s = W q + 1/µ 2) Ave. # in line = # of arrivals during an ave. wait in line: L q = λw q Little s Formula L s = λw s Also holds for the system EX: Suppose λ = 10 customers/hour & W q = ½ hour. Then L q = customers 6
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