Chapter 11

1. Chapter 11 Queuing Models

2. A Basic Queueing System

3. 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

4. Examples of Commercial Service SystemsThat Are Queueing Systems

5. Examples of Internal Service SystemsThat Are Queueing Systems

6. Examples of Transportation Service SystemsThat Are Queueing Systems

7. 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?

8. 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?

9. 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.

10. 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.

11. The Queue • 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 discipline refers 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.

12. Number of Servers • Single Server • Multiple Servers

13. Arrivals • The time between consecutive arrivals to a queueing system are called the interarrival times. • The expected number of arrivals per unit time is referred to as the mean arrival rate. • The symbol used for the mean arrival rate is l = Mean arrival rate for customers coming to the queueing system where l is the Greek letter lambda. • The mean of the probability distribution of interarrival times is 1 / l = Expected interarrival time • Most queueing models assume that the form of the probability distribution of interarrival times is an exponential distribution.

14. 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.

15. Service • 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 queueing models assume that the service time has a particular probability distribution. • The symbol used for the mean of the service time distribution is 1 / m = Expected service timewhere m is the Greek letter mu. • The interpretation of m itself is the mean service rate.m = Expected service completions per unit time for a single busy server

16. 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. • 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.

17. 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.

18. Properties of the Exponential Distribution • 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 queueing 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 lack-of-memory property.

19. Notation • Parameters: l = customer arrival ratem = service rate (1/m = average service time)s = number of servers • Performance Measures Lq = average number of customers in the queueL = average number of customers in the systemWq= average waiting time in the queueW = average waiting time (including service)Pn= probability of having n customers in the systemr = system utilization

20. 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.

21. Herr Cutter’s 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. • The table shows his queueing system in action over a typical morning.

22. Evolution of the Number of Customers

23. Labels for Queueing Models To identify which probability distribution is being assumed for service times (and for interarrival times), a queueing model conventionally is labeled as follows: Distribution of service times — / — / — Number of Servers Distribution of interarrival times The symbols used for the possible distributions areM = Exponential distribution (Markovian)D = Degenerate distribution (constant times)GI = General independent interarrival-time distribution (any distribution)G = General service-time distribution (any arbitrary distribution)

24. Summary of Usual Model Assumptions • Interarrival times are independent and identically distributed according to a specified probability distribution. • All arriving customers enter the queueing system and remain there until service has been completed. • The queueing 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. • The queueing 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.

25. Choosing a Measure of Performance • Managers who oversee queueing systems are mainly concerned with two measures of performance: • How many customers typically are waiting in the queueing system? • How long do these customers typically have to wait? • When customers are internal to the organization, the first measure tends to be more important. • Having such customers wait causes lost productivity. • 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.

26. Defining the Measures of Performance 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 queueing system is in a steady-state condition.

27. Relationship between L, W, Lq, and Wq • Since 1/m is the expected service timeW = Wq+ 1/m • Little’s formula states thatL = lWandLq= lWq • Combining the above relationships leads toL = Lq + l/m

28. The Dupit Corp. Problem • The Dupit Corporation is a longtime leader in the office photocopier marketplace. • Dupit’s service division is responsible for providing support to the customers by promptly repairing the machines when needed. This is done by the company’s service technical representatives, or tech reps. • Current policy: Each tech rep’s 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. • Proposed New Service Standard: The average waiting time before a tech rep begins the trip to the customer site should not exceed two hours.

29. Alternative Approaches to the 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. • 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 one-person tech rep territories by larger territories served by multiple tech reps. • Approach Suggested by the Vice President for Marketing: Give owners of the new printer-copier priority for receiving repairs over the company’s other customers.

30. The Queueing System for Each Tech Rep • 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.)

31. Notation for Single-Server Queueing Models • l = Mean arrival rate for customers = Expected number of arrivals per unit time1/l = expected interarrival time • m = Mean service rate (for a continuously busy server) = Expected number of service completions per unit time1/m = expected service time • r = the utilizationfactor= the average fraction of time that a server is busy serving customers = l / m

32. The M/M/1 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. • The queueing system has one server. • The expected number of customers in the system is L = r / (1 –r) = l / (m– l) • The expected waiting time in the system is W = (1 / l)L = 1 / (m – l) • The expected waiting time in the queue is Wq= W – 1/m = l / [m(m – l)] • The expected number of customers in the queue is Lq= lWq = l2 / [m(m – l)] = r2 / (1 – r)

33. The M/M/1 Model • Theprobability of having exactly n customers in the system is Pn = (1 – r)rnThus,P0 = 1 – rP1 = (1 – r)rP2 = (1 – r)r2 : : • The probability that the waiting time in the system exceeds t is P(W > t) = e–m(1–r)t for t ≥ 0 • The probability that the waiting time in the queue exceeds t is P(Wq > t) = re–m(1–r)t for t ≥ 0

34. M/M/1 Queueing Model for the Dupit’s Current Policy

35. John Phixitt’s 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). • John Phixitt’s suggested approach is to lower the tech rep’s utilization factor sufficiently to meet the new service requirement. Lower r = l / m, until Wq≤ 1/4 day,wherel = (Number of machines assigned to tech rep) / 50.

36. M/M/1Model for John Phixitt’s Suggested Approach(Reduce Machines/Rep)

37. The M/G/1 Model • Assumptions • Interarrival times have an exponential distribution with a mean of 1/l. • Service times can have any probability distribution. You only need the mean (1/m) and standard deviation (s). • The queueing system has one server. • The probability of zero customers in the system is P0 = 1 – r • The expected number of customers in the queue is Lq= [l2s2 + r2] / [2(1 – r)] • The expected number of customers in the system is L = Lq + r • The expected waiting time in the queue is Wq= Lq/ l • The expected waiting time in the system is W = Wq+ 1/m

38. The Values of s and Lqfor the M/G/1 Modelwith Various Service-Time Distributions

39. 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). • 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. • 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.

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

41. 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). • With multiple servers, the formula for the utilization factor becomesr = l / smbut still represents that average fraction of time that individual servers are busy.

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

43. CFO Suggested Approach (Combine Into Teams) • The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq≤ 1/4 day). • 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. • A territory with two tech reps: • Number of machines = 300 (versus 150 before) • Mean arrival rate = l = 6 (versus l = 3 before) • Mean service rate = m = 4 (as before) • Number of servers = s = 2 (versus s = 1 before) • Utilization factor = r = l/sm = 0.75 (as before)

44. M/M/s Model for the CFO’s Suggested Approach(Combine Into Teams of Two)

45. M/M/s Model for the CFO’s Suggested Approach(Combine Into Teams of Three)

46. Comparison of Wq with Territories of Different Sizes

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

48. Priority Queueing 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. • After deferring to higher priority customers, the customers within each priority class are served on a first-come-fist-served basis. • 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. • Preemptive priorities: The lowest priority customer being served is preempted (ejected back into the queue) whenever a higher priority customer enters the queueing system.

49. Preemptive Priorities Queueing 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. • All service times have an exponential distribution with a mean of 1/m, regardless of the priority class involved. • The queueing system has a single server. • The utilization factor for the server is r = (l1 + l2 + … + ln) / m

50. Nonpreemptive Priorities Queueing 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. • All service times have an exponential distribution with a mean of 1/m, regardless of the priority class involved. • The queueing system can have any number of servers. • The utilization factor for the servers is r = (l1 + l2 + … + ln) / sm