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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Where is There Waiting? Service Facility –Fast-food restaurants –Post office –Grocery store –Bank Disneyland Highway traffic Manufacturing Equipment awaiting repair Phone or computer network Product orders

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Why is There Waiting? Example #1: McDonalds –50 customers arrive per hour –Service rate is 60 customers per hour Example #2: Doctor’s Office –Arrivals are scheduled to arrive every 20 minutes. –The doctor spends an average of 18 minutes with each patient.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., System Characteristics Number of servers Arrival and service pattern –rate of arrivals and service –distribution of arrivals and service Maximum size of the queue Queue disciplince –FCFS? –Priority system? Population size –Infinite or finite?

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Measures of System Performance Average number of customers waiting –in the system –in the queue Average time customers wait –in the system –in the queue Which measure is the most important?

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Number of Servers Single Server Multiple Servers

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Arrival Pattern A Poisson distribution is usually assumed. A good approximation of random arrivals. Lack-of-memory property: Probability of an arrival in the next instant is constant, regardless of the past.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Service Pattern Either an exponential distribution is assumed, –Implies that the service is usually short, but occasionally long –If service time is exponential then service rate is Poisson –Lack-of-memory property: The probability that a service ends in the next instant is constant (regardless of how long its already gone). –Decent approximation if the jobs to be done are random. –Not a good approximation if the jobs to be done are always the same. Or any distribution –Only single-server model is easily solved.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Maximum Size of Queue Most queueing models assume an infinite queue length is possible. If the queue length is limited, a finite queue model can be used.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Queue Discipline Most queueing systems assume customers are served first-come first-served. If certain customers are given priority, a priority queueing model can be used. –Nonpreemptive: Finish customer in service before taking a new one. –Preemptive: If priority customer arrives, any regular customer in service is preempted (put back in the queue).

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Population Source Most queueing models assume an infinite population source. If the number of potential customers is small, a finite source model can be used. –Number in system affects arrival rate (fewer potential arrivals when more in system) –Okay to assume infinite if N > 20.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Models 1.Single server, exponential service time (M/M/1) 2.Single server, general service time (M/G/1) 3.Multiple servers, exponential service time (M/M/s) 4.Finite queue (M/M/s/K) 5.Priority queue (nonpreemptive and preemptive) 6.Finite calling population A Taxonomy — / — / — (and an optional fourth element / —) ArrivalServiceNumber ofMaximum DistributionDistributionServersin Queue where M = Exponential (Markovian) D = deterministic (constant) G = general distribution

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Notation Parameters: = customer arrival rate = service rate (1/ = average service time) s= number of servers Performance Measures L q = average number of customers in the queue L= average number of customers in the system W q = average waiting time in the queue W= average waiting time (including service) P n = probability of having n customers in the system = system utilization

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Model 1 (M/M/1) Customers arrive to a small-town post office at an average rate of 10 per hour (Poisson distribution). There is only one postal employee on duty and he can serve customers in an average of 5 minutes (exponential distribution).

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Model 2 (M/G/1) ABC Car Wash is an automated car wash. Each customer deposits four quarters in a coin slot, drives the car into the auto-washer, and waits while the car is automatically washed. Cars arrive at an average rate of 20 cars per hour (Poisson). The service time is exactly 2 minutes.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Model 3 (M/M/s) A grocery store has three registers open. Customers arrive to check out at an average of 1 per minute (Poisson). The service time averages 2 minutes (exponential).

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Model 4 (M/M/s/K) A call center that handles the tech support for a software manufacturer currently has 10 telephone lines, with three people fielding the calls. Customers call at an average rate of 40 per hour (Poisson). A customer can be served in an average of four minutes (exponential).

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Model 5a (Nonpreemptive Priority Queue) Consider a small-town hospital emergency room (ER) that has just one doctor on duty. When patients arrive, they are classified as either critical or non-critical. When the doctor is finished treating a patient, she takes the next critical patient. If there are no critical patients, then she takes the next non-critical patient. The ER doctor spends an average of 10 minutes (exponential) treating each patient before they are either released or admitted to the hospital. An average of 1 critical patient and 3 non-critical patients arrive each hour (Poisson).

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Model 5b (Preemptive Priority Queue) Reconsider the same small-town hospital emergency room (ER). Now suppose they change their policy so that if a critical patient arrives while a non-critical patient is being treated, the doctor stops treating the non-critical patient, and immediately starts treating the critical patient. Only when there are no critical patients to be treated does the doctor start treating non-critical patients.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Model 6 (Finite Calling Population) Consider a PC-Board assembly facility. There are six automated component insertion machines. Unfortunately, they are very prone to break down. Each operating machine breaks down every eight hours or so (exponential distribution). Because these machines are so prone to break down, a full-time repairperson is kept on staff just to repair these machines. Each repair takes an average of one hour (exponential distribution). On average, how many machines are operating at a time?

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Application of Queueing Models We can use the results from queueing models to make the following types of decisions: –How many servers to employ. –How large should the waiting space be. –Whether to use a single fast server or a number of slower servers. –Whether to have a general purpose server or faster specific servers.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Total Cost The goal is to minimize total cost = cost of servers + cost of waiting

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Example #1: How Many Servers? The MIS department of a high tech company handles employee requests for assistance when computer questions arise. Employees requiring assistance phone the MIS department with their questions (but may have to wait on hold if all of the tech support staff are busy). The MIS department receives an average of 40 requests for assistance per hour (Poisson). The average question can be answered in 3 minutes (exponential). The MIS staff is paid an average of $15 per hour. The average employee earns $25 per hour. Question: What is the optimal size of the MIS tech support staff?

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Example #1: How Many Servers

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., A Data Table for Example #1: How Many Servers?

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Example #2: How Many Servers? A McDonalds franchise is trying to decide how many registers to have open during their busiest time, the lunch hour. Customers arrive during the lunch hour at a rate of 98 customers per hour (Poisson distribution). Each service takes an average of 3 minutes (exponential distribution). Question #1: If management would not like the average customer to wait longer than five minutes in line, how many registers should they open? Question #2: If management would like no more than 5% of customers to wait more than 5 minutes, how many registers should they open?

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Example #2: How Many Servers?

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Example #3: How Much Waiting Space? A photo development shop operates a drive-through lane where customers can drop off film to be developed and pick up developed photos. Customers arrive at an average rate of 40 per hour (Poisson). Each service takes an average of 1 minute (exponential). They are remodeling the parking area and drive-through lane. They would like the drive-through lane to hold all of the customers at least 95% of the time. Question: How many cars must the drive-through lane be able to hold?

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Example #3: How Much Waiting Space?

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Example #4: One Fast Server or Many Slow Servers A McDonalds is considering changing the way that they serve customers. Customers arrive at an average rate of 50 per hour. Current System: For most of the day (all but their lunch hour), they have three registers open. Each cashier takes the customer’s order, collects the money, and then gets the burgers and pours the drinks. This takes an average of 3 minutes per customer (exponential distribution). Proposed System: They are considering having just one cash register. While one person takes the order and collects the money, another will pour the drinks, and another will get the burgers (like Wendys). The three together think they can serve a customer in an average of 1 minute. Question: Should they switch to the proposed system?

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Slow Servers (McDonalds) 1 Fast Server (Wendys)

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Example #5: General or Specific Servers A small bank in a mall has two tellers. The bank handles two kinds of customers: merchant customers and regular customers. Each arrive at an average rate of 20 customers per hour (for a total arrival rate of 40 customers per hour). Current System (Specific Servers): Currently one teller handles only merchant customers and one teller handles only regular customers. The service time for both tellers averages 2 minutes (exponential). Proposed System (General Servers): The bank manager is considering changing the setup to allow each teller to handle both merchant customers and regular customers. Since the tellers would have to handle both types of jobs, their efficiency would decrease to a mean service time of 2.2 minutes. Question: Should they switch to the proposed system?

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Current (Specific Servers) Proposed (General Servers)

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., LL Bean LL Bean’s mail order business –Mail order phone lines open 24 hours per day, 365 days per year –78,000 calls per week (average) –Seasonal variations as well as variability during each day How LL Bean estimates the number of servers needed –Each of the week’s 168 hours in a week is modeled separately as a period to be staffed –Each hour modeled as an M/M/s queue –Arrival rates and service rates estimated from historical data –Service standard: no more than 15% of calls wait more than 20 seconds –Full-time, part-time, and temporary workers scheduled to meet service standard

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