Chapter 14. Queuing Models. 第十四章. 排队模型. 运筹学 排队模型. 现实生活中的排队模型. Bank Airport Hospital Road Manufacturing Hotel Restaurant WC ……. 运筹学 排队模型. 如何以最经济的方式控制排队系统，使其达到特定的要求？ 提供过多的服务能力来控制排队系统将会造成过量的成本 提供的服务能力不足将会导致过多的等待，降低顾客满意度并造成顾客流失，减少收益. 排队模型存在的问题. 什么是排队论. 什么是 排队论 ?. 什么是排队论.
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Chapter 14. Queuing Models
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Elements of a Queuing Model (Section 14.1)()
Some Examples of Queuing Systems (Section 14.2)()
Measures of Performance for Queuing Systems (Section 14.3)()
A Case Study: The Dupit Corp. Problem (Section 14.4)()
Some SingleServer Queuing Models (Section 14.5)()
Some MultipleServer Queuing Models (Section 14.6)()
Priority Queuing Models (Section 14.7)()
Some Insights about Designing Queuing Systems (Section 14.8)()
Economic Analysis of the Number of Servers to Provide (Section 14.9)()
There is a high likelihood of small interarrival times, but a small chance of a very large interarrival time. This is characteristic of interarrival times in practice. ()
For most queuing systems, the servers have no control over when customers will arrive. Customers generally arrive randomly. ()
Having random arrivals means that interarrival times are completely unpredictable, in the sense that the chance of an arrival in the next minute is always just the same. ()
The only probability distribution with this property of random arrivals is the exponential distribution. ()
The fact that the probability of an arrival in the next minute is completely uninfluenced by when the last arrival occurred is called the lackofmemory property. ()
The number of customers in the queue (or queue size) is the number of customers waiting for service to begin. ()
The number of customers in the system is the number in the queue plus the number currently being served. ()
The queue capacity is the maximum number of customers that can be held in the queue. ()
An infinite queue is one in which, for all practical purposes, an unlimited number of customers can be held there. ()
When the capacity is small enough that it needs to be taken into account, then the queue is called a finite queue. ()
The queue disciplinerefers to the order in which members of the queue are selected to begin service. ()
The most common is firstcome, firstserved (FCFS). ()
Other possibilities include random selection, some priority procedure, or even lastcome, firstserved. ()
When a customer enters service, the elapsed time from the beginning to the end of the service is referred to as the service time. ()
Basic queuing models assume that the service time has a particular probability distribution. ()
The symbol used for the mean of the service time distribution is ()
Interarrival times are independent and identically distributed according to a specified probability distribution. ()
All arriving customers enter the queuing system and remain there until service has been completed. ()
The queuing system has a single infinite queue, so that the queue will hold an unlimited number of customers (for all practical purposes). ([])
The queue discipline is firstcome, firstserved. ()
The queuing system has a specified number of servers, where each server is capable of serving any of the customers. ()
Each customer is served individually by any one of the servers. ()
Service times are independent and identically distributed according to a specified probability distribution. ()
L = Expected number of customers in the system, including those being served (the symbol L comes from Line Length).
Lq= Expected number of customers in the queue, which excludes customers being served.
W = Expected waiting time in the system (including service time) for an individual customer (the symbol W comes from Waiting time).
Wq = Expected waiting time in the queue (excludes service time) for an individual customer.
These definitions assume that the queuing system is in a steadystate condition. ()
In addition to knowing what happens on the average, we may also be interested in worstcase scenarios. ()
What will be the maximum number of customers in the system? (Exceeded no more than, say, 5% of the time.)(95%?)
What will be the maximum waiting time of customers in the system? (Exceeded no more than, say, 5% of the time.)(95%?)
Statistics that are helpful to answer these types of questions are available for some queuing systems ():
Pn = Steadystate probability of having exactly n customers in the system.(n)
P(W t) = Probability the time spent in the system will be no more than t.(t)
P(Wq t) = Probability the wait time will be no more than t.(t)
Examples of common goals():
No more than three customers 95% of the time: P0 + P1 + P2 + P3 0.95 (95%3)
No more than 5% of customers wait more than 2 hours: P(W 2 hours) 0.95 (25%)
The Dupit Corporation is a longtime leader in the office photocopier marketplace.
Dupits service division is responsible for providing support to the customers by promptly repairing the machines when needed. This is done by the companys service technical representatives, or tech reps.
Current policy: Each tech reps territory is assigned enough machines so that the tech rep will be active repairing machines (or traveling to the site) 75% of the time.
A repair call averages 2 hours, so this corresponds to 3 repair calls per day.
Machines average 50 workdays between repairs, so assign 150 machines per rep.
75%()
23
50150

Proposed New Service Standard: The average waiting time before a tech rep begins the trip to the customer site should not exceed two hours. ()
Approach Suggested by John Phixitt: Modify the current policy by decreasing the percentage of time that tech reps are expected to be repairing machines. ()
Approach Suggested by the Vice President for Engineering: Provide new equipment to tech reps that would reduce the time required for repairs. ()
Approach Suggested by the Chief Financial Officer: Replace the current oneperson tech rep territories by larger territories served by multiple tech reps. ()
Approach Suggested by the Vice President for Marketing: Give owners of the new printercopier priority for receiving repairs over the companys other customers. ()
The customers: The machines needing repair.()
Customer arrivals: The calls to the tech rep requesting repairs.()
The queue: The machines waiting for repair to begin at their sites.()
The server: The tech rep.()
Service time: The total time the tech rep is tied up with a machine, either traveling to the machine site or repairing the machine. (Thus, a machine is viewed as leaving the queue and entering service when the tech rep begins the trip to the machine site.)
()
Assumptions()
Interarrival times have an exponential distribution with a mean of .( )
Service times have an exponential distribution with a mean of 1/m. (1/m )
The queuing system has one server. ()
Theprobability of having exactly n customers in the system is (n)
The probability that the waiting time in the system exceeds t is (t)
The probability that the waiting time in the queue exceeds t is (t)
150100
100006/
50003/
The probability of zero customers in the system is ()
25()
With multiple servers, the formula for the utilization factor becomes ()
but still represents that average fraction of time that individual servers are busy. ()
The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq 1/4 day). (2)
The Chief Financial Officer has suggested combining the current oneperson tech rep territories into larger territories that would be served jointly by multiple tech reps. ()
CFO Suggested Approach (Teams of Three)
The utilization factor for the server is ()
The utilization factor for the servers is ()
The mean arrival rates for the two classes of copiers are ()
The proportion of printercopiers is expected to increase, so in a couple years ()
Decision: Adopt fourth proposal (except for sparsely populated areas where second proposal should be adopted). ([])
When designing a singleserver queuing system, beware that giving a relatively high utilization factor (workload) to the server provides surprisingly poor performance for the system. ([])
Decreasing the variability of service times (without any change in the mean) improves the performance of a queuing system substantially. ([][])
Multipleserver queuing systems can perform satisfactorily with somewhat higher utilization factors than can singleserver queuing systems. For example, pooling servers by combining separate singleserver queuing systems into one multipleserver queuing system greatly improves the measures of performance.
Applying priorities when selecting customers to begin service can greatly improve the measures of performance for highpriority customers. ()
In many cases, the consequences of making customers wait can be expressed as a waiting cost. ()
The manager is interested in minimizing the total cost. ()
TC = Expected total cost per unit time
()
SC = Expected service cost per unit time
()
WC = Expected waiting cost per unit time
()
The objective is then to choose the number of servers so as to ()
Minimize TC = SC + WC
When each server costs the same (Cs= cost of server per unit time), ()SC = Cs s
When the waiting cost is proportional to the amount of waiting (Cw = waiting cost per unit time for each customer), ()WC = Cw L
The Acme Machine Shop has a tool crib for storing tool required by shop mechanics. ()
Two clerks run the tool crib. ()
The estimates of the mean arrival rate l and the mean service rate (per server) m are ( [] )l = 120 customers per hour (120)m = 80 customers per hour (80)
The total cost to the company of each tool crib clerk is $20/hour, so Cs= $20. (20)
While mechanics are busy, their value to Acme is $48/hour, so Cw = $48. (48)
Choose s so as to Minimize TC = $20s + $48L. (TC)
SIMULATION
The end of chapter 14