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Chapter 14. Queuing Models

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Chapter 14. Queuing Models

.


  • Bank

  • Airport

  • Hospital

  • Road

  • Manufacturing

  • Hotel

  • Restaurant

  • WC



?



A Basic Queuing System




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)()

Table of Contents ()


A Case Study: The Dupit Corp. Problem (Section 14.4)()

Some Single-Server Queuing Models (Section 14.5)()

Some Multiple-Server Queuing Models (Section 14.6)()

Table of Contents ()


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)()

Table of Contents ()


Herr Cutters Barber Shop

  • Herr Cutter is a German barber who runs a one-man barber shop. ()

  • Herr Cutter opens his shop at 8:00 A.M. (8)

  • The table shows his queuing system in action over a typical morning. ()


Herr Cutters Barber Shop


Arrivals()

  • The time between consecutive arrivals to a queuing system are called the interarrival times. ()

  • The expected number of arrivals per unit time is referred to as the mean arrival rate. ()


Arrivals()

  • The symbol used for the mean arrival rate is ()

  • The mean of the probability distribution of interarrival times is ()


Arrivals()

  • Most queuing models assume that the form of the probability distribution of interarrival times is an exponential distribution. ()


Evolution of the Number of Customers


The Exponential Distribution for Interarrival Times


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. ()

Properties of the Exponential Distribution




For most queuing systems, the servers have no control over when customers will arrive. Customers generally arrive randomly. ()

Properties of the Exponential Distribution


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. ()

Properties of the Exponential Distribution


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 lack-of-memory property. ()

Properties of the Exponential Distribution


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 ()


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. ()

The Queue ()


When the capacity is small enough that it needs to be taken into account, then the queue is called a finite queue. ()

The Queue ()


The queue disciplinerefers to the order in which members of the queue are selected to begin service. ()

The most common is first-come, first-served (FCFS). ()

Other possibilities include random selection, some priority procedure, or even last-come, first-served. ()

The Queue ()


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. ()

Service ()


The symbol used for the mean of the service time distribution is ()

  • The interpretation of itself is the mean service rate. ( )

  • = Expected service completions per unit time for a single busy server ()

Service ()


Some Service-Time Distributions

  • Exponential Distribution ()

    • The most popular choice. ()

    • Much easier to analyze than any other. ()

    • Although it provides a good fit for interarrival times, this is much less true for service times. ()


Some Service-Time Distributions

  • Provides a better fit when the service provided is random than if it involves a fixed set of tasks. ()

  • Standard deviation: s = Mean ()

  • Constant Service Times ()

    • A better fit for systems that involve a fixed set of tasks. ()

    • Standard deviation (): s = 0.


  • Some Service-Time Distributions

    • Erlang Distribution ()

      • Fills the middle ground between the exponential distribution and constant. ()

      • Has a shape parameter, k that determines the standard deviation. (k)



    Erlang Distribution ()


    Standard Deviation and Mean for Distributions


    Labels for Queuing Models

    • To identify which probability distribution is being assumed for service times (and for interarrival times), a queuing model conventionally is labeled as follows: ([])


    Labels for Queuing Models

    • The symbols used for the possible distributions are ()

      • M = Exponential distribution (Markovian) ()

      • D = Degenerate distribution (constant times) ()


    Labels for Queuing Models

    • Ek = Erlang distribution (shape parameter = k) ([=k])

    • GI = General independent interarrival-time distribution (any distribution) ([])

    • G = General service-time distribution (any arbitrary distribution) ([])


    Summary of Usual Model Assumptions

    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. ()


    Summary of Usual Model Assumptions

    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 first-come, first-served. ()


    Summary of Usual Model Assumptions

    The queuing system has a specified number of servers, where each server is capable of serving any of the customers. ()


    Summary of Usual Model Assumptions

    Each customer is served individually by any one of the servers. ()

    Service times are independent and identically distributed according to a specified probability distribution. ()


    Examples of Commercial Service Systems that Are Queuing Systems


    Examples of Internal Service Systems That Are Queuing Systems


    Examples of Transportation Service Systems That Are Queuing Systems


    Choosing a Measure of Performance

    • Managers who oversee queuing systems are mainly concerned with two measures of performance: ()

      • How many customers typically are waiting in the queuing system? ()

      • How long do these customers typically have to wait? ()


    Choosing a Measure of Performance

    • When customers are internal to the organization, the first measure tends to be more important. ([])

      • Having such customers wait causes lost productivity. ()


    Choosing a Measure of Performance

    • Commercial service systems tend to place greater importance on the second measure. ([])

      • Outside customers are typically more concerned with how long they have to wait than with how many customers are there. ()


    Defining the Measures of Performance

    L = Expected number of customers in the system, including those being served (the symbol L comes from Line Length).


    Defining the Measures of Performance

    Lq= Expected number of customers in the queue, which excludes customers being served.


    Defining the Measures of Performance

    W = Expected waiting time in the system (including service time) for an individual customer (the symbol W comes from Waiting time).


    Defining the Measures of Performance

    Wq = Expected waiting time in the queue (excludes service time) for an individual customer.


    Defining the Measures of Performance

    These definitions assume that the queuing system is in a steady-state condition. ()


    Relationship between L, W, Lq and Wq


    In addition to knowing what happens on the average, we may also be interested in worst-case scenarios. ()

    What will be the maximum number of customers in the system? (Exceeded no more than, say, 5% of the time.)(95%?)

    Using Probabilities as Measures of Performance


    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 ():

    Using Probabilities as Measures of Performance


    Pn = Steady-state 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)

    Using Probabilities as Measures of Performance


    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%)

    Using Probabilities as Measures of Performance


    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.

    The Dupit Corp. Problem


    The Dupit Corp. Problem


    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.

    The Dupit Corp. Problem


    75%()

    23

    50150

    The Dupit Corp. Problem


    -


    Proposed New Service Standard: The average waiting time before a tech rep begins the trip to the customer site should not exceed two hours. ()

    The Dupit Corp. Problem


    Approach Suggested by John Phixitt: Modify the current policy by decreasing the percentage of time that tech reps are expected to be repairing machines. ()

    Alternative Approaches to the Problem


    Approach Suggested by the Vice President for Engineering: Provide new equipment to tech reps that would reduce the time required for repairs. ()

    Alternative Approaches to the Problem


    Approach Suggested by the Chief Financial Officer: Replace the current one-person tech rep territories by larger territories served by multiple tech reps. ()

    Alternative Approaches to the Problem


    Approach Suggested by the Vice President for Marketing: Give owners of the new printer-copier priority for receiving repairs over the companys other customers. (-)

    Alternative Approaches to the Problem


    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.()

    The Queuing System for Each 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.)

    The Queuing System for Each Tech Rep


    ()

    The Queuing System for Each Tech Rep


    Notation for Single-Server Queuing Models


    Notation for Single-Server Queuing Models


    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. ()

    The M/M/1 Model


    The M/M/1 Model


    The M/M/1 Model


    The M/M/1 Model

    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 M/M/1 Model

    The probability that the waiting time in the queue exceeds t is (t)


    M/M/1 Queuing Model for the Dupits Current Policy


    John Phixitts Approach (Reduce Machines/Rep)

    • The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq 1/4 day). (2)


    John Phixitts Approach (Reduce Machines/Rep)

    • John Phixitts suggested approach is to lower the tech reps utilization factor sufficiently to meet the new service requirement. ()


    John Phixitts Approach (Reduce Machines/Rep)


    M/M/1 Model for John Phixitts Suggested Approach (Reduce Machines/Rep)


    150100

    100006/

    50003/

    1


    The M/G/1 Model

    • Assumptions()

    • Interarrival times have an exponential distribution with a mean of 1/l. (1/l )

    • Service times can have any probability distribution. You only need the mean (1/m) and standard deviation( s). ()

    • The queuing system has one server.()


    The probability of zero customers in the system is ()

    The M/G/1 Model

    • The expected number of customers in the queue is ()


    The M/G/1 Model

    • The expected number of customers in the system is ()

    • The expected waiting time in the queue is ()


    The M/G/1 Model

    • The expected waiting time in the system is ()



    VP for Engineering Approach (New Equipment)

    • The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq 1/4 day).(2)


    VP for Engineering Approach (New Equipment)

    • The Vice President for Engineering has suggested providing tech reps with new state-of-the-art equipment that would reduce the time required for the longer repairs. ()


    VP for Engineering Approach (New Equipment)

    • After gathering more information, they estimate the new equipment would have the following effect on the service-time distribution: ()

      • Decrease the mean from 1/4 day to 1/5 day. ()

      • Decrease the standard deviation from 1/4 day to 1/10 day. ()


    M/G/1 Model for the VP of Engineering Approach (New Equipment)

    25()


    The M/M/s Model

    • Assumptions ()

      • Interarrival times have an exponential distribution with a mean of 1/l. ( )

      • Service times have an exponential distribution with a mean of 1/m. ( )

      • Any number of servers (denoted by s). (s)


    With multiple servers, the formula for the utilization factor becomes ()

    The M/M/s Model

    but still represents that average fraction of time that individual servers are busy. ()


    Values of L for the M/M/s Model for Various Values of s


    The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq 1/4 day). (2)

    CFO Suggested Approach (Combine Into Teams)


    The Chief Financial Officer has suggested combining the current one-person tech rep territories into larger territories that would be served jointly by multiple tech reps. ()

    CFO Suggested Approach (Combine Into Teams)


    CFO Suggested Approach (Combine Into Teams)


    M/M/s Model for the CFOs Suggested Approach (Combine Into Teams of Two)


    CFO Suggested Approach (Teams of Three)


    M/M/s Model for the CFOs Suggested Approach (Combine Into Teams of Three)


    Comparison of Wq with Territories of Different Sizes


    3


    Values of L for the M/D/s Model for Various Values of s


    Values of L for the M/Ek/2 Model for Various Values of k


    Priority Queuing Models

    • General Assumptions:

      • There are two or more categories of customers. Each category is assigned to a priority class. Customers in priority class 1 are given priority over customers in priority class 2. Priority class 2 has priority over priority class 3, etc.


    Priority Queuing Models

      • 1223


    Priority Queuing Models

    • General Assumptions():

      • After deferring to higher priority customers, the customers within each priority class are served on a first-come-fist-served basis. ()


    Priority Queuing Models

    • Two types of priorities

      • Nonpreemptive priorities: Once a server has begun serving a customer, the service must be completed (even if a higher priority customer arrives). However, once service is completed, priorities are applied to select the next one to begin service.


    Priority Queuing Models


    Priority Queuing Models

    • Two types of priorities

      • Preemptive priorities: The lowest priority customer being served is preempted (ejected back into the queue) whenever a higher priority customer enters the queuing system.


    Priority Queuing Models

      • ()


    Preemptive Priorities Queuing Model

    • Additional Assumptions ()

      • Preemptive priorities are used as previously described.()

      • For priority class i (i = 1, 2, , n), the interarrival times of the customers in that class have an exponential distribution with a mean of 1/li. (i )


    Preemptive Priorities Queuing Model

    • Additional Assumptions ()

      • All service times have an exponential distribution with a mean of 1/m, regardless of the priority class involved. ( )

      • The queuing system has a single server. ()


    Preemptive Priorities Queuing Model

    The utilization factor for the server is ()


    Nonpreemptive Priorities Queuing Model

    • Additional Assumptions ()

      • Nonpreemptive priorities are used as previously described. ()

      • For priority class i (i = 1, 2, , n), the interarrival times of the customers in that class have an exponential distribution with a mean of 1/li. (i )


    Nonpreemptive Priorities Queuing Model

    • Additional Assumptions ()

      • All service times have an exponential distribution with a mean of 1/m, regardless of the priority class involved. ( )

      • The queuing system can have any number of servers. ()


    Nonpreemptive Priorities Queuing Model

    The utilization factor for the servers is ()


    VP of Marketing Approach (Priority for New Copiers)

    • The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq 1/4 day).

    • The Vice President of Marketing has proposed giving the printer-copiers priority over other machines for receiving service. The rationale for this proposal is that the printer-copier performs so many vital functions that its owners cannot tolerate being without it as long as other machines.


    VP of Marketing Approach (Priority for New Copiers)

    • 2

    • --


    VP of Marketing Approach (Priority for New Copiers)

    The mean arrival rates for the two classes of copiers are ()

    The proportion of printer-copiers is expected to increase, so in a couple years (-)


    Nonpreemptive Priorities Model for VP of Marketings Approach (Current Arrival Rates)


    Nonpreemptive Priorities Model for VP of Marketings Approach (Future Arrival Rates)


    Expected Waiting Times with Nonpreemptive Priorities


    4


    The Four Approaches Under Considerations


    The Four Approaches Under Considerations

    Decision: Adopt fourth proposal (except for sparsely populated areas where second proposal should be adopted). ([])


    When designing a single-server queuing system, beware that giving a relatively high utilization factor (workload) to the server provides surprisingly poor performance for the system. ([])

    Some Insights About Designing Queuing Systems


    Decreasing the variability of service times (without any change in the mean) improves the performance of a queuing system substantially. ([][])

    Some Insights About Designing Queuing Systems


    Multiple-server queuing systems can perform satisfactorily with somewhat higher utilization factors than can single-server queuing systems. For example, pooling servers by combining separate single-server queuing systems into one multiple-server queuing system greatly improves the measures of performance.

    Some Insights About Designing Queuing Systems


    Some Insights About Designing Queuing Systems


    Applying priorities when selecting customers to begin service can greatly improve the measures of performance for high-priority customers. ()

    Some Insights About Designing Queuing Systems


    Effect of High-Utilization Factors (Insight 1)


    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. ()

    Economic Analysis of the Number of Servers to Provide


    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

    Economic Analysis of the Number of Servers to Provide


    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

    Economic Analysis of the Number of Servers to Provide


    The Acme Machine Shop has a tool crib for storing tool required by shop mechanics. ()

    Two clerks run the tool crib. ()

    Acme Machine Shop


    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)

    Acme Machine Shop


    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)

    Acme Machine Shop


    Excel Template for Choosing the Number of Servers


    SIMULATION


    The end of chapter 14


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