In addition, two new appendices have been added,discussing transforms and generating functions as well as thefundamentals of differential and difference equations.
New examplesare now included along with problems that incorporate QtsPlussoftware, which is freely available via the book's related Website. With its accessible style and wealth of real-world examples,Fundamentals of Queueing Theory, Fourth Edition is an idealbook for courses on queueing theory at the upper-undergraduate andgraduate levels.
It is also a valuable resource for researchers andpractitioners who analyze congestion in the fields oftelecommunications, transportation, aviation, and managementscience.
It features Excel and Quattro software that allows greater flexibility in the understanding of the nature, sensitivities and responses of waiting- line systems to parameter and environmental changes. Its excellent organizational structure allows quick reference to specific models and its clear presentation coupled with the use of the QTS software solidifies the understanding of the concepts being presented.
Its excellent organizational structure allows quick reference to specific models and its clear presentation. This update takes a numerical approach to understanding and making probable estimations relating to queues, with a comprehensive outline of simple and more advanced queueing models.
Newly featured topics of the Fourth Edition include: Retrial queues Approximations for queueing networks Numerical inversion of transforms Determining the appropriate number of servers to balance quality and cost of service Each chapter provides a self-contained presentation of key concepts and formulae, allowing readers to work with each section independently, while a summary table at the end of the book outlines the types of queues that have been discussed and their results.
In addition, two new appendices have been added, discussing transforms and generating functions as well as the fundamentals of differential and difference equations.
New examples are now included along with problems that incorporate QtsPlus software, which is freely available via the book's related Web site. With its accessible style and wealth of real-world examples, Fundamentals of Queueing Theory, Fourth Edition is an ideal book for courses on queueing theory at the upper-undergraduate and graduate levels. It is also a valuable resource for researchers and practitioners who analyze congestion in the fields of telecommunications, transportation, aviation, and management science.
This is an Erlang type-n distribution see Section 3. From the last line, we see that Wq t represents the mixture of a discrete random variable and a continuous random variable. Specifically, the term on the right is the CDF of an exponential random variable, with parameter p, 1 - p , weighted by the factor p. The term on the left 1 - p is the CDF of a random variable that takes on the constant value of zero, weighted by the factor 1- p. Thus Wq t is the mixture of an exponential distribution and a discrete probability mass at zero.
Wq t can be further simplified to yield a more convenient form: 2. In a similar manner, the CDF of the total time in system can be derived. Let T denote the total time an arriving customer spends in the system in steady state. It can be shown Problem 2. The derivation of 2. The CDFs, Wet and Wq t , are discipline-dependent, as can be seen in their derivations by the requirement that a customer's waiting time is determined by the amount of time it takes to serve the customers found upon arrival.
Finally, we emphasize that the measures of effectiveness used here are all calculated for steady state. These results are not applicable to the initial few customers who arrive soon after opening. Also, the results do not apply when the arrival rate or service rate varies in time, as is the case, for example, when there is a rush of customers at a particular time of day.
Using 2. In a similar manner, the distribution of the total time in the system including service can be calculated using 2. Like the M I Mil queue, this queue can be modeled as a birth -death process Figure 2. In contrast, the rate of service completions or "deaths " depends on the number in the system. If there are c or more customers in the system, then all c servers must be busy. Since each server processes customers with rate p" the combined service-completion rate for the system is cp,.
Thus we get L W ! We now obtain Wq O , the probability that a customer has zero delay in queue before receiving service. Equivalently, 1 - Wq O is the probability that a customer has nonzero delay in queue before receiving service. This measure of congestion is often used in managing call centers.
In such systems, an arriving caller waits "on hold" in a virtual queue when all servers are busy. In queue, the caller typically hears music or informational announcements. To evaluate L rn In! This formula gives the probability that an arriving customer is delayed in the queue i. In particular, the model ignores complexities such as abandonments, retrials, and nonstationary arrivals - factors that may be important in modeling call centers. Also, the model assumes an infinite queue size.
For call centers, this corresponds to an infinite number of available access lines into the center. The time for a support person to serve one customer is exponentially distributed with a mean of 5 minutes.
The support center has 3 technical staff to assist callers. What is the probability that a customer is able to immediately access a support staff, without being delayed on hold?
Assume that customers do not abandon their calls. Since C c, r represents the probability of positive delay, the probability of no delay is. How many servers are needed? We now obtain the complete probability distributions of the waiting times, W t and Wq t , in a manner similar to that of Section 2.
Putting this result together with 2. As with the MIMl1 queue, Wq t is the mixture of a discrete probability mass at zero and an exponential distribution.
Similar statements would be true for the other M 1MI c measures of effectiveness. We leave as an exercise Problem 2.
To find the formula for the CDF of the system waiting time, we first split the situation into two separate possibilities, namely, those customers having no wait in queue [occurring with probability Wq 0 ] and those customers having a positive wait in queue [occurring with probability 1 - Wq O ]. The time in the system for the first class of customers is just the time in service, since there is no wait in the queue.
For these customers, the CDF of the time in the system is identical to the CDF of the time in service, an exponential distribution with mean lip. For the second class of customers, the time in the system is the sum of the wait in queue plus the time in service.
This convolution can also be written as the difference of the two exponential functions see Problem 2. There are three ophthalmologists on duty. A test takes, on average, 20 min, and the actual time is found to be approximately exponentially distributed around this average.
Clients arrive according to a Poisson process with a mean of 6th, and patients are taken on a first-come, first-served basis. The hospital planners are interested in knowing 1 the average number of people waiting; 2 the average amount of time a patient spends at the clinic; and 3 the average percentage idle time of each of the doctors. Thus we wish to calculate L q , W, and the percentage idle time of a server. We begin by calculating Po, since this factor appears in all the formulas derived for the measures of effectiveness.
Next, we have already shown see Table 1. A larger number of servers improves quality of service to the customers but incurs a higher cost to the queue owner. The problem is to find the number of servers that adequately balances the quality and cost of service.
The approximation works for queues with a large number of servers. Before discussing the approximation, we first observe that in steady state the number of servers must be greater than the offered load r. Otherwise, the queue is unstable.
Thus the problem of choosing the number of servers c is equivalent to choosing the number of servers fl in excess of the offered load. Suppose that the owner of the queue has observed the system for a long time and is satisfied with its overall performance, considering both the congestion experienced by the customers and the cost of paying the servers.
How many new servers should the owner hire? There are several lines of reasoning that can be taken to answer this question.
Choose c to maintain approximately a constant traffic intensity p. It may seem reasonable to keep this ratio constant. Choose c to maintain approximately a constant measure of congestion.
An example congestion measure is 1 - Wq O , the probability that a customer is delayed in the queue, which can be obtained from the Erlang-C formula 2. The number of servers can be found by increasing c until 1 - Wq 0 ::; a or by using a binary search on c. Choose c to maintain approximately a constant "padding" of servers.
These three approaches have sometimes been called the "quality domain," the "quality and efficiency domain" QED , and the "efficiency domain" Gans et al. Table 2. Quality domain 2. Quality and efficiency domain 3. The number of servers is held approximately proportional to the offered load. Yet, as the offered load increases, the probability of delay in the queue decreases. In other words, large queueing systems that have been scaled using the quality-domain approach have very little queueing delay.
In the efficiency domain, there is an emphasis on minimizing cost at the expense of service quality. Here, the number of excess servers is held approximately fixed.
As the offered load increases, congestion in the queue increases. In other words, large queueing systems that have been scaled using the efficiency-domain approach are nearly always congested. The QED provides a balance between the quality and efficiency domains. In this domain, the objective is to maintain a fixed quality of service. That is, 1 - Wq O is held approximately constant as the system grows. In contrast, in the quality domain 1 - Wq O goes to 0; in the efficiency domain 1 - Wq O goes to 1.
One difficulty with this approach is that the choice for c is not a simple formula. In particular, c is chosen according to 2. The main result of this section is that although there is no simple formula for choosing c in the QED, there is a simple approximate formula. The basic idea is that the number of excess servers should increase with the square root of the offered load. Specifically, the solution to 2. Intuitively, 2.
In the preceding example, T increased by a factor of 4. Thus to maintain the same quality of service,. The square-root law 2. In summary, the sequence of queues has approximately the same quality of service provided that the excess number of servers grows with the square root of the offered load. The square-root law can be used in a relative sense without specifying the precise values of these constants. Alternatively, the constants can be used to approximate absolute levels of service.
Specifically, to choose the number of servers c that achieves a quality of service a, first find 3 that satisfies 2. A similar approximation is to let 3 equal the 1 - a quantile of the standard normal distribution Kolesar and Green, We first compute the exact value of c so that the probability of nonzero delay in 2. Alternatively, using the approximation, the value of 3 that satisfies 2. Then from 2.
Figure 2. The individual points are the exact values found from the inversion problem in 2. The solid lines are approximate values obtained from the square-root law in 2. As seen, the approximation works quite well for these examples. In Section 6. It then follows from 2. However, for the finite-waiting-space case, we need to adjust this result and Little's formula as well , since a fraction PK of the arrivals do not join the system, because they have come when there is no waiting space left.
Thus the actual rate of arrivals to join the system must be adjusted accordingly. Since Poisson arrivals see time averages the PASTA property , it follows that the effective arrival rate seen by the servers is.. We henceforth denote any such adjusted input rate as..
We know that the quantity r 1 - PK must be less than e, since the average number of customers in service must be less than the total number of available servers. Expected values for waiting times can readily be obtained by use of Little's formula as 2. The derivation of the waiting-time CDF is somewhat complicated, since the series are finite, although they can be expressed in terms of cumulative Poisson sums, as we shall show.
Finally, to get the CDF Wq t for the line delays, we note, in a fashion similar to the derivation leading to 2. It is reasonable to assume that cars wait in such a way that when a stall becomes vacant, the car at the head of the line pulls up to it. The station can accommodate at most four cars waiting seven in the station at one time.
The arrival pattern is Poisson with a mean of one car every minute during the peak periods. Fussy, the chief inspector, wishes to know the average number in the system during peak periods, the average wait including service , and the expected number per hour that cannot enter the station because of full capacity.
We first calculate Po from 2. To find the average wait during peak periods, 2. This might suggest an alternative setup for the inspection station. This stationary distribution can be obtained from 2. This is the probability of a full system at any time in steady state.
The original physical situation that motivated Erlang to devise this model was the simple telephone network.
Incoming calls arrive as a Poisson stream, service times are mutually independent exponential random variables, and all calls that arrive finding every trunk line busy i. The model has always been of great value in telecommunications design. But the great importance of this formula lies in the very surprising fact that 2. That is, the steady-state system probabilities are only a function of the mean service time, not of the underlying CDF.
Erlang was also able to deduce the formula for the case when service times are constant. While this result was later shown to be correct, his proof was not quite valid. Later works by Vaulot , Pollaczek , Palm , Kosten , and others smoothed out Erlang's proof and supplied proofs for the general service-time distribution.
A further addition to this sequence of papers was work by Takacs , which supplied some additional results for the problem. We shall prove the validity of Erlang's loss formula for general service in Chapter 5, Section 5. In particular, when the number of servers c is large, terms like e! For example, ! Applications like call centers often require values of c greater than This section gives alternate formulas that are more practical to implement on a computer.
In addition, the alternate formulas for computing B c, r can be used to compute measures of congestion for the M 1MIe queue, such as the Erlang-C formula 2. For the Erlang-B formula, it can be shown Problem 2. This method avoids numerical overflow that may be encountered with direct application of 2. Then, C c, r can be written as a function of B c, r as follows Problem 2.
For example, 2. This can be calculated iteratively using 2. Alternatively, B 4, 2 can be calculated from 2.
The average number in queue Lq can be calculated using 2. This model is often referred to as the ampleserver problem.
A self-service situation is a good example of the use of such a model. We make use of the general birth -death results with. A and J. The value of. L is not restricted in any way for the existence of a steady-state solution. It also turns out we show this in Section 5. That is, Pn depends only on the mean service time and not on the form of the service-time distribution. It is not surprising that this is true here in light of a similar result we mentioned previously for M I M lei c, since Pn of 2.
The expected system size is the mean of the Poisson distribution of 2. L, and the waiting-time distribution function Wet is identical to the service-time distribution, namely, exponential with mean 1 I J. It has found from past surveys that people turning on their television sets on Saturday evening during prime time can be described rather well by a Poisson distribution with a mean of IOO,OOOIh.
Surveys have also shown that the average person tunes in for 90 min and that viewing times are approximately exponentially distributed. We now treat a problem where the calling population is finite of size M, and future event occurrence probabilities are functions of the system state.
A typical application of this model is that of machine repair, where the calling population is the machines, an arrival corresponds to a machine breakdown, and the repair technicians are the servers. Because of these assumptions, we can use the birth-death theory developed previously. For the machine repair problem, "number in system" corresponds to the number of broken machines. Abate and Whitt point out that computing values with 2. We can show, using properties of the Bessel functions, that 2.
However, since transient solutions require solving sets of differential equations, numerical methods can often be successfully employed. We treat this topic in some detail in Chapter 8, Section 8.
A busy period begins when a customer arrives at an idle channel and ends when the channel next becomes idle. A busy cycle is the sum of a busy period and an adjacent idle period, or equivalently, the time between two successive departures leaving an empty system, or two successive arrivals to an empty system.
Therefore the CDF of the busy period is sufficient to describe the busy cycle also and is found as follows. Now the Laplace transform of Po t , Po s , is the first coefficient of the power series P z, s and is thus found as P O,s. It is not too difficult to extend the notion of the busy period conceptually to the multichannel case. Recall that for one channel a busy period is defined to begin with the arrival of a customer at an idle channel and to end when the channel next becomes idle.
Then it should be clear that Pi-1 t will, in fact, be the required CDF, and its derivative the density. You are told that a small single-server, birth-death-type queue with finite capacity cannot hold more than three customers. The finite-capacity constraint of Problem 2. Derive W t and w t the total-waiting-time CDF and its density as given by the equations 2. The approach is to plot the cumulative count of arrivals on the same graph as the cumulative count of departures. Then it can be seen that the area between these two step functions from the beginning of a busy period to the beginning of the next a busy cycle is the accumulated total of the system waiting times of all the customers who have entered into the system during this busy cycle.
Use this argument to derive an empirical version of Little's formula over a busy cycle. A graduate research assistant "moonlights" in the food court in the student union in the evenings. He is the only one on duty at the counter during the hours he works.
Each customer is served one at a time and the service time is thought to follow an exponential distribution with a mean of 4 min. Answer the following questions. If he can grade 22 papers an hour on average when working continuously, how many papers per hour can he average while working his shift? A rent-a-car maintenance facility has capabilities for routine maintenance oil change, lubrication, minor tune-up, wash, etc.
Cars arrive there according to a Poisson process at a mean rate of three per day, and service time to perform this maintenance seems to have an exponential distribution with a mean of :]4 day.
This also increases their operating costs. Up to what value can the operating cost increase before it is no longer economically attractive to make the change? The machine can paint one part at a time. The cost of owning and operating the painting machine is strictly a function of its speed. Determine the value of J. L that minimizes the cost of the painting operation. So it does not require that all waiting customers form a single line, and instead they make every arrival randomly choose one of three lines formed before each server during the weekday lunch period.
But they are so traditional about managing their lines that barriers have been placed between the lines to prevent jockeying. Suppose that the overall stream of incoming customers has settled in at a constant rate of 60th Poisson-distributed and that the time to complete a customer's order is well described by an exponential distribution of mean seconds, independent and identically from one customer to the next.
Assuming steady state, what is the average total system size? What is the expected steady-state system size now? To celebrate the event, the campus book store ordered tee shirts. On the day of the sale, demand for shirts was steady throughout the day and fairly well described by a Poisson process with a rate of 66 per hour. There were four cash registers in operation and the average time of a transaction was 3. Service times were approximately exponentially distributed.
You are the owner of a small book store. You have two cash registers. Customers wait in a single line to purchase books at one of the two registers. The time to complete the purchase transactions for one customer follows an exponential distribution with mean 3 minutes. What is the hourly rate that you make money?
Now, what is the hourly rate that you make money? It has established 25 investigation teams to analyze and evaluate each accident or incident to make sure it is properly reported to accident investigation boards.
Each of these teams is dispatched to the locale of the accident or incident as each requirement for such support occurs. Support is only rendered those commands that have neither the facilities nor qualified personnel to conduct such services. Each accident or incident will require a team being dispatched for a random amount of time, apparently exponential with mean of 3 weeks.
At any given time, two teams are not available due to personnel leaves, sickness, and so on. Find the expected time spent by an accident or incident in and waiting for evaluation. An organization is presently involved in the establishment of a telecommunication center so that it may provide a more rapid outgoing message capability.
Overall, the center is responsible for the transmission of outgoing messages and receives and distributes incoming messages.
The center manager at this time is primarily concerned with determining the number of transmitting personnel required at the new center. Outgoing message transmitters are responsible for making minor corrections to messages, assigning numbers when absent from original message forms, maintaining an index of codes and a day file of outgoing messages, and actually transmitting the messages.
All outgoing messages will be processed in the order they are received and follow a Poisson process with a mean rate of 21 per 7-h day. Processing on messages requiring transmission must be started within an average of 2 h from the time they arrive at the center.
Determine the minimum number of transmitting personnel to accomplish this service criterion. If the service criterion were to require the probability of any message waiting for the start of processing for more than 3 h to be less than.
A small branch bank has two tellers, one for receipts and one for withdrawals. Customers arrive to each teller's cage according to a Poisson distribution with a mean of h. The total mean arrival rate at the bank is h. The service time of each teller is exponential with a mean of 2 min. However, since the tellers would have to handle both receipts and withdrawals, their efficiency would decrease to a mean service time of 2.
Compare the present system with the proposed system with respect to the total expected number of people in the bank, the expected time a customer would have to spend in the bank, the probability of a customer having to wait more than 5 min, and the average idle time of the tellers. The Hott Too Trott Heating and Air Conditioning Company must choose between operating two types of service shops for maintaining its trucks.
It estimates that trucks will arrive at the maintenance facility according to a Poisson distribution with mean rate of one every 40 min and believes that this rate is independent of which facility is chosen. In the first type of shop, there are dual facilities operating in parallel; each facility can service a truck in 30 min on average the service time follows an exponential distribution.
In the second type there is a single facility, but it can service a truck in 15 min on average service times are also exponential in this case. To help management decide, they ask their operations research analyst to answer the following questions: a How many trucks, on average, will be in each of the two types of facilities?
They also know from previous experience in running dual-facility shops that the cost of operating such a facility is one dollar per minute including labor, overhead, etc. What would the operating cost per minute have to be for operating the single-facility shop in order for there to be no difference between the two types of shops?
The ComPewter Company, which leases out high-end computer workstations, considers it necessary to overhaul its equipment once a year. The maintenance time for a machine has an exponential distribution with a mean of 6 h.
In this case, the maintenance time for a machine has an exponential distribution with a mean of 3 h. For both alternatives, the machines arrive according to a Poisson input with a mean arrival rate of one every 8 h since the company leases such a large number of machines, we can consider the machine population as infinite.
Which alternative should the company choose? For Problem 2. A small drive-it-through-yourself car wash, in which the next car cannot go through the washing procedure until the car in front is completely finished, has a capacity to hold on its grounds a maximum of 10 cars including the one in wash.
The company has found its arrivals to be Poisson with mean rate of 20 carsth, and its service times to be exponential with a mean of 12 min. What is the average number of cars lost to the firm every lO-h day as a result of its capacity limitations? Under the assumption that customers will not wait if no seats are available, Example 2.
Her shop is open on Saturdays from A. This office can seat an additional four people. Should Cutt rent? The Fowler-Heir Oil Company operates a crude-oil unloading port at its major refinery. The port has six unloading berths and four unloading crews. When all berths are full, arriving ships are diverted to an overflow facility 20 miles down river. Tankers arrive according to a Poisson process with a mean of one every 2 h.
It takes an unloading crew, on average, 10 h to unload a tanker, the unloading time following an exponential distribution. Tankers waiting for unloading crews are served on a first-come, first-served basis. Company management wishes to know the following: a On average, how many tankers are at the port?
Assume that construction and maintenance costs would amount to X dollars per year. The company estimates that to divert a tanker to the overflow port when the main port is full costs Y dollars.
What is the relation between X and Y for which it would pay for the company to build an extra berth at the main port? Fly-Bynite Airlines has a telephone exchange with three lines, each manned by a clerk during its busy periods.
During their peak three hours per h period, many callers are unable to get into the exchange there is no provision for callers to hold if all servers are busy. We may assume that the number of people not getting through during off-peak hours is negligible. The three peak hours occur during the 8-h day shift. At all other times, one clerk can handle all the traffic, and since the company never closes the exchange, exactly one clerk is used on the off shifts.
Assume that the cost of adding lines to the exchange is negligible. A call center has 24 phone lines and 3 customer service representatives. The time to process each call is exponential with a mean of 10 minutes. If all of the service representatives are busy, an arriving the phone lines. If all of the customer is placed on hold, but ties up on phone lines are tied up, the customer receives a busy signal and the call is lost.
Fixing the number of service representatives, what is the optimal number of phone lines you should have? Prove the iterative relationship in 2. Prove the relationship in 2. The Good Writers Correspondence Academy offers a go-at-your-own-pace correspondence course in good writing.
New applications are accepted at any time, and the applicant can enroll immediately. An applicant's mean completion time is found to be 10 weeks, with the distribution of completion times being exponential. On average, how many pupils are enrolled in the school at any given time? A manufacturer of a very expensive, rather infrequently demanded item uses the following inventory control procedure.
She keeps a safety stock of S units on hand. The customer demand for units can be described by a Poisson process with mean A. Every time a request for a unit is made a customer demand , an order is placed at the factory to manufacture another this is called a one-for-one ordering policy.
If p z could be determined, one could optimize E[G] with respect to S. Hence relate p z to Pn. State explicitly what the input and service mechanisms are.
Farecard machines that dispense tickets for riding on the subway have a mean operating time to breakdown of 45 h. It takes a technician on average 4 h to repair a machine, and there is one technician at each station. Assume that the time to breakdown and the time to repair are exponentially distributed. What is the number of installed machines necessary to assure that the probability of having at least five operational is greater than.
Show for the basic machine repair model no spares that qn M , the failure arrival point probabilities for a population of size M, equal Pn M - 1 , the general-time probabilities for a population of size M - 1. Derive qn M given by 2. While that is no proof, the statement can be shown to hold in general see Sevick and Mitrani, , or Lavenberg and Reiser, A coin-operated dry-cleaning store has five machines.
The operating characteristics of the machines are such that any machine breaks down according to a Poisson process with mean breakdown rate of one per day. A repairman can fix a machine according to an exponential distribution with a mean repair time of one-half day. Currently, three repairmen are on duty. The manager, Lew Cendirt, has the option of replacing these three repairmen with a super-repairman whose salary is equal to the total of the three regulars, but who can fix a machine in one-third the time, that is, in one-sixth day.
Should he be hired? Suppose that each of five machines in a given shop breaks down according to a Poisson law at an average rate of one every 10 h, and the failures are repaired one at a time by two maintenance people operating as two channels, such that each machine has an exponentially distributed servicing requirement of mean 5 h.
Very often in real-life modeling, even when the calling population is finite, an infinite-source model is used as an approximation. To compare the two models, calculate L for Example 2. How do you think p affects the approximation? Find the average operating costs per hour of Example 2. What now is the best policy? Assume we have a two-state, state-dependent service model as described in Section 2. Suppose that the customers are lawntreating machines owned by the Green Thumb Lawn Service Company and these machines require, at random times, greasing on the company's twospeed greasing machine.
What is the optimal switch point k? Derive the steady-state system-size probabilities for a single-server model with Poisson input and exponential state-dependent service with mean rates f-ll 1 ; n 2. Suppose that customers balk at joining the queue when it is too long. Determine the steady-state probability that there are n in the system. Suppose that the M I Mil reneging model of Section 2. Find the stationary system-size distribution. Derive the steady-state M I M solution directly from the transient.
Use the properties of Laplace transforms to find the functions whose Laplace transforms are the following: a b c d 2. Show that the moment generating function of the sum of independent random variables is equal to the product of their moment generating functions. Use the result of Problem 2. That is, we allow changes of more than one over infinitesimal time intervals but insist on retaining the memoryless Markovian property.
The Chapman - Kolmogorov and backward and forward equations, plus the resultant balance equations, are all still valid, and together are the essence of the approach to solution for these nonbirth-death Markovian problems. We now recall the discussions of Section 1. Figure 3. For an arbitrary batch-size distribution X, a general set of rate balance equations can be derived Problem 3.
To solve the system of equations given by 3. Difference-equation methods are often used instead to solve the problem when the maximum batch is small. Under the proper parameter settings, the processes treated in this chapter meet the conditions of Theorem 1. Multiplying each equation of 3. After the first stage many items are found to have one or more defects, which must be repaired before they enter the second stage.
It is the job of one worker to make the necessary adjustments to put the assembly back into the stream of the process. The number of defects per item is registered automatically, and it exceeds two an extremely small number of times. But it is subsequently noted that the rates of defects have increased, although not continuously. It is therefore decided to put another person on the job, who will concentrate on repairing those units with two defects, while the original worker works only on singles.
When to add the additional person will be decided on the basis of a cost analysis. Now there are a number of alternative cost structures available, and it is decided by management that the expected cost of the system to the company will be based on the average delay time of assemblies in for repair, which is directly proportional to the average number of units in the system, L.
If a second repairer now sets up a separate service channel, the additional cost of his or her time is incurred, over and above the cost of the items in the queue. In this case, we have two queues. The expected number of required repairs in the system is then the sum of the expected values of the two streams. Using our values for the parameters gives a decision criterion of C 2 3.
Like with all preceding editions, this latest update of the classic textbook features a unique combination of the theoretical and timely real-world applications. Highlighting chapter-end exercises and problems—all of which have been classroom-tested and improved by the authors in advanced undergraduate and graduate-level courses— Fundamentals of Queueing Theory, 5th Edition is an ideal textbook for courses in applied mathematics, probability and statistics, queueing theory and stochastic processes.
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