Queueing Theory/Waiting Line: Models and Analysis Navneet Vidyarthi

2 A Simple Queuing (Waiting Line) System Population Source:  Infinite Source: Supermarkets, Banks, Restaurants,…  Finite Source: Nurse, Doctors

3 Queuing System – No. of waiting lines  Single Waiting Line  E.g. airline counters, rental car counters, restaurants, amusement park attractions, call centers.  The advantage is the customer’s perception of fairness in terms of equitable waits. That is, the customer is not penalized by picking the slow line but is served in a true first-come, first- served fashion.  The single line approach eliminates jockeying behavior.  Finally, a single-line, multiple-server system has better performance in terms of waiting times than the same system with a line for each server.

4 Queuing System – No. of waiting lines  Multiple Waiting Lines  The multiple-line configuration is appropriate when specialized servers are used or when space considerations make a single line inconvenient.  For example, in a grocery store some registers are express lanes for customers with a small number of items.  Using express lines reduces the waiting time for customers making smaller purchases or who paid premium (Business class passengers).

5 Queuing System – No. of Servers  Single-server systems  Small retail stores with a single checkout counter,  A theater with a single person selling tickets and controlling admission into the show, or  A ballroom with a single person controlling admission.  Multiserver systems  Have parallel service providers offering the same service.  Multiserver examples include grocery stores (multiple cashiers), drive-through banks (multiple drive-through windows), and gas stations (multiple gas pumps).

6 Queuing System –Server Arrangement  Services require a single activity or a series of activities and are identified by the term “ phase”.  In a single-phase system, the service is completed all at once, such as with a bank transaction or a grocery store checkout.  In a multiphase system, the service is completed in a series of steps, such as at a fast-food restaurant with ordering, pay, and pick-up windows; or many manufacturing processes.

7 Some Examples

8 Queuing System –Arrival and Service Patterns  Arrival rate specifies the average number of customers per time period.  For example, a system may have two customers arrive on average each hour.

9  Service rate specifies the average number of customers that can be serviced during a time period.

10 Queuing System –Arrival and Service Patterns  If the number of customers you can serve per time period is less than the average number of customers arriving, the waiting line grows infinitely. You never catch up with the demand!

11  Waiting line models usually assume that  customers arrive according to a Poisson probability distribution  The Poisson distribution specifies the probability that a certain number of customers will arrive in a given time period (such as per hour).

12  Waiting line models usually assume that  service times are described by an exponential distribution  The exponential distribution describes the service times as the probability that a particular service time will be less than or equal to a given amount of time.

13  In practice, we conduct chi-square goodness-of-fit test to verify the assumptions on the distributions of arrival times and the service times. Problem Solving Tip: Make sure the arrival rate and service rate are for the same time period, that is, the number of customers per hour, or per day, or per week.

14 Waiting Line Priority Rules

15  A waiting line priority rule determines which customer is served next.  A frequently used priority rule is first-come, first-served.  Generally customers consider first-come, first-served to be the fairest method for determining priority.  Other rules include  best customers first,  highest profit customer first,  quickest service requirement first,  largest service requirement first,  emergencies first, and so on.

16 Some terms often used…  Balking: The customer decides not to enter the waiting line.  Reneging: The customer enters the line but decides to exit before being served.  Jockeying: The customer enters one line and then switches to a different line in an effort to reduce the waiting time.

17 Performance Measure  The average number of customers waiting in line.  The average number of customers in system.  The average time customers spend waiting.  The average time a customer spends in the system.  The system utilization rate.  Probability that an arriving customer will have to wait for service or will have to wait more than a specified length of time.

18 Waiting line models in this course Model 1: Single-server, Exponential service durations Model 2: Single-server, Constant service durations

19 Basic relationships that always holds Server Utilization (ρ): ratio of demand to capacity. In waiting line models, this means ratio of the average arrival rate (λ) to the average service rate (Mµ).  λ = mean arrival rate of customers (average number of customers arriving per unit of time)  µ = mean service rate (average number of customers that can be served per unit of time)  M = Number of servers

20 Average number of customer in service (r) = Average number of customer in system (L s ) = Average number of customer in queue (L q )+ Average number of customer in service (r).

21 Average time spent by the customer in service = ? Average time spent by the customer in system (W s ) = Average time spent in queue (W q )+ Average time spent in service (1/μ).

22  Relation between L and W

23 Model 1: Single-server, Exponential service durations  Assumptions:  The customers are patient (no balking, reneging, or jockeying) and come from a population that can be considered infinite.  Customer arrivals are described by a Poisson distribution with a mean arrival rate of λ This means that the time between successive customer arrivals follows an exponential distribution with an average of 1/ λ.  The customer service rate is described by a Poisson distribution with a mean service rate of (mu). This means that the service time for one customer follows an exponential distribution with an average of 1/μ.  The waiting line priority rule used is first-come, first-served.

24 Example (model 1) The computer lab at a University has a help desk to assist students working on computer spreadsheet assignments. The students patiently form a single line in front of the desk to wait for help.  Students are served based on a first-come, first-served priority rule.  On average, 15 students per hour arrive at the help desk. Student arrivals are best described using a Poisson distribution.  The help desk server can help an average of 20 students per hour, with the service rate being described by an exponential distribution. Calculate the following operating characteristics of the service system. (a) The average utilization of the help desk server (b) The average number of students waiting in line (c) The average number of students in service (d) The average number of students in the system

25 (e) The average time a student spends waiting in line (f) The average time a student spends in service (f) The average time a student spends in the system (g) The probability that the system is busy (h) The probability that the system is empty (this is same the probability of zero customers in the system) (i) The probability of having four students in the system (j) The probability of having fewer than four students in the system (j) The probability of having more than four students in the system The help desk has a policy that states that every student who spends 10 min or more waiting in the queue receives $1 gift coupon. Determine how many gift coupon should they be expecting to giving away in a week period (assume the desk opens for 12 hours/day).

26 First thing to do: Check for the Stability of the Queue !!!  The first step is to check for stability of the queue before using any formula –  because all the formulas that we discuss in this chapter is useful only when the queue is stable  else the result will make no sense.  The queue is stable when  the service rate (μ) is more than the arrival rate (λ) (this implies the capacity of system is more than the demand)

27 In the above example,  service rate, μ = 20 students per hour  arrival rate, λ = 15 students per hour  The queue is stable as μ > λ, hence proceed ahead.  Assume that there is a single server at the help desk.  Hence, use the single-server, exponential service duration model (a) The average utilization of the help desk server

28 (b) The average number of students waiting in line (c) The average number of students in service (d) The average number of students in the system

29 (e) The average time a student spends waiting in line (f) The average time a student spends in service (g) The average time a student spends in the system Alternatively, we could have used the formula L = λW, and got the same answer as follows (just a check!!!):

30 (g) The probability that the system is busy (this is same as the system utilization) (h) The probability that the system is empty (= the probability of zero customers in the system) (= the probability or fraction of time that the help desk is idle) Note that the system is either busy serving customer or idle,

31 (i) The probability of having four students in the system (j) The probability of having fewer than four students in the system (k) The probability of having more than four students in the system The probability that there are more than four students in the system equals one minus the probability that there are four or fewer students in the system:

32  The help desk has a policy that states that every student who spends 10 min or more waiting in the queue receives $1 gift coupon. Determine how many gift coupon should they be expecting to give away in one week period (if the desk runs 24 hours a day, 7 days a week).  The average time a student spends waiting in line = 9 min.  However the actual waiting time of some of the students might exceed the 10 min threshold.  Let us find the probability that a student waits at least 10 min in a queue,  Note that almost one in every three student spends 10 min or more waiting in the queue.  Hence, number of gift coupon they should be expecting to give away per day (if the desk opens 12 hours a day) = *___ = or $ 59 coupons per day.

33 Model 2: Single-server, Constant service durations  Assumptions:  The customers are patient (no balking, reneging, or jockeying) and come from a population that can be considered infinite.  Customer arrivals are described by a Poisson distribution with a mean arrival rate of λ. This means that the time between successive customer arrivals follows an exponential distribution with an average of 1/ λ.  The customer service rate is constant with a service rate of µ or the service time for one customer is exactly 1/μ units.  The waiting line priority rule used is first-come, first-served.

34 Example (model 2) A service station has an automatic car wash with three-minute operation with single bay. On a typical Saturday morning, cars arrive at a mean rate of 15 per hours, with the arrivals tending to follow a Poisson distribution. Calculate the following operating characteristics of the service station: (a) The average utilization of the system (b) The average number of cars waiting in line (c) The average number of cars in service (d) The average number of cars in the system (e) The average time a car spends waiting in line (f) The average time a car spends in service (f) The average time a car spends in the system (g) The probability that the car wash is busy (h) The probability that the car wash is empty

35 First thing to do: Check for the Stability of the Queue !!!  The first step is to check for stability of the queue before using any formula or else the result will make no sense.  The queue is stable when  the service rate (μ) is more than the arrival rate (λ) (this implies the capacity of system is more than the demand)  In the above example,  service rate, μ = 20 cars per hour  arrival rate, λ = 15 cars per hour  The queue is stable as μ > λ, hence proceed ahead.  Use the single-server, constant service duration model (model 2)

36 (a) The average utilization of the system (b) The average number of cars waiting in line How does this compare with the Lq of the help desk example? (c) The average number of cars in service (d) The average number of cars in the system

37 (e) The average time a car spends waiting in line (f) The average time a student spends in service (g) The average time a car spends in the system Alternatively, we could have used the formula L = λW, and got the same answer as follows (just a check!!!):

38 (g) The probability that the car wash is busy (this is same as the system utilization) (h) The probability that the car wash is empty (= the probability of zero customers in the system) (= the probability or fraction of time that the help desk is idle) Note that the system is either busy serving customer or idle,

39 Answers  Check for stability of system: Stable system, because Mμ (=3*18) > λ (=45), hence proceed.  (a) The average utilization of the help desk  (b) The probability that there are no students in the system Using the lookup table for P 0 corresponding to M =3 and system utilization = 0.83, we get or 4.6%.

40  (c) The average number of students waiting in line  (d) The average number of students in service  (e) The average number of students in the system = (3.5 students)

41  (f) The average time a student spends waiting in line  (g) The average time a student spends in service  (h) The average time a student spends in the system (0.134 hrs)