Queueing Models

A Basic Queueing System Figure 14.1 A basic queueing system, where each customer is indicated by C and each server by S. Although this figure shows four servers, some queueing systems (including the example in this section) have only a single server.

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. Customer Time of Arrival Haicut Begins Duration of Haircut Haircut Ends 1 8:03 17 minutes 8:20 2 8:15 21 minutes 8:41 3 8:25 19 minutes 9:00 4 8:30 15 minutes 9:15 5 9:05 20 minutes 9:35 6 9:43 — Table 14.1 The data for Herr Cutter’s first five customers.

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 a Poisson and Service an Exponential distribution

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.

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 time where 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

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 are M = Exponential distribution (Markovian) D = Degenerate distribution (constant times) Ek = Erlang distribution (shape parameter = 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 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.

Examples of Commercial Service Systems That Are Queueing Systems Type of System Customers Server(s) Barber shop People Barber Bank teller services Teller ATM machine service ATM machine Checkout at a store Checkout clerk Plumbing services Clogged pipes Plumber Ticket window at a movie theater Cashier Check-in counter at an airport Airline agent Brokerage service Stock broker Gas station Cars Pump Call center for ordering goods Telephone agent Call center for technical assistance Technical representative Travel agency Travel agent Automobile repair shop Car owners Mechanic Vending services Vending machine Dental services Dentist Roofing Services Roofs Roofer Table 14.3 Examples of commercial service systems that are queueing systems.

Examples of Internal Service Systems That Are Queueing Systems Type of System Customers Server(s) Secretarial services Employees Secretary Copying services Copy machine Computer programming services Programmer Mainframe computer Computer First-aid center Nurse Faxing services Fax machine Materials-handling system Loads Materials-handling unit Maintenance system Machines Repair crew Inspection station Items Inspector Production system Jobs Machine Semiautomatic machines Operator Tool crib Machine operators Clerk Table 14.4 Examples of internal service systems that are queueing systems.

Examples of Transportation Service Systems That Are Queueing Systems Type of System Customers Server(s) Highway tollbooth Cars Cashier Truck loading dock Trucks Loading crew Port unloading area Ships Unloading crew Airplanes waiting to take off Airplanes Runway Airplanes waiting to land Airline service People Airplane Taxicab service Taxicab Elevator service Elevator Fire department Fires Fire truck Parking lot Parking space Ambulance service Ambulance Table 14.5 Examples of transportation service systems that are queueing systems.

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.

Relationship between L, W, Lq, and Wq Since 1/m is the expected service time W = Wq + 1/m Little’s formula states that L = lW and Lq = lWq Combining the above relationships leads to L = Lq + l/m

Using Probabilities as Measures of Performance 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.) What will be the maximum waiting time of customers in the system? (Exceeded no more than, say, 5% of the time.) Statistics that are helpful to answer these types of questions are available for some queueing systems: Pn = Steady-state probability of having exactly n customers in the system. P(W ≤ t) = Probability the time spent in the system will be no more than t. P(Wq ≤ t) = Probability the wait time will be no more than t. Examples of common goals: No more than three customers 95% of the time: P0 + P1 + P2 + P3 ≥ 0.95 No more than 5% of customers wait more than 2 hours: P(W ≤ 2 hours) ≥ 0.95

Notation for Single-Server Queueing Models l = Mean arrival rate for customers = Expected number of arrivals per unit time 1/l = expected interarrival time m = Mean service rate (for a continuously busy server) = Expected number of service completions per unit time 1/m = expected service time r = the utilization factor = the average fraction of time that a server is busy serving customers = l / m

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)

The M/M/1 Model The probability of having exactly n customers in the system is Pn = (1 – r)rn Thus, P0 = 1 – r P1 = (1 – r)r P2 = (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

Why is There Waiting? Example #1: McDonalds 50 customers arrive per hour Service rate is 60 customers per hour Example #2: Doctor’s Office Arrivals are scheduled to arrive every 20 minutes. The doctor spends an average of 18 minutes with each patient. For each example, ask if there will be waiting. For example #1, they can serve customers more quickly than they arrive. Will there still by waiting? Yes, because of variability in arrivals. Arrivals tend to come in bunches (e.g., a bus full of little-leaguers after a game, or a whole large family) causing lines to form. Also, 50 customers per hour may be an average. During the busy lunch hour, the arrival rate may be higher, and lines will form. For example #2, there is little to no variability in arrivals. Will there still be waiting? Yes, because of variability in service. A really long service (i.e., for a very sick patient) will cause the following appointments to be delayed. The doctor being called away on an emergency will cause the appointments to be delayed.

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?

Measures of System Performance Average number of customers waiting in the system in the queue Average time customers wait Which measure is the most important? 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.

Number of Servers Single Server Multiple Servers Discuss examples of each: Single Server -- Wendys Multiple server (one line): bank, airport Multiple server (line for each server): McDonalds, grocery store These two multiple-server setups are identical from a modeling perspective. The average line lengths and waiting times will be identical. However, the left one is more fair, because it always is FCFS. In the right one, a customer can get “unlucky” and get in a slow line, and be served after someone who arrived after them but got in a quicker line. Studies have shown that waiting does not bother customers so much as perceived unfairness does. Something else to discuss: What’s better, a single-server with many people working together to make the service go fast (the model at Wendys), vs. multiple servers, with one person at each that does the whole service more slowly (the model at McDonalds). We will return to examine this question in the following lecture.

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.

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.

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. Examples with a finite queue: call center (telephone lines limited), drive through lane.

Queue Discipline Most queueing systems assume customers are served first-come first-served. If certain customers are given priority, a priority queueing model can be used. Nonpreemptive: Finish customer in service before taking a new one. Preemptive: If priority customer arrives, any regular customer in service is preempted (put back in the queue).

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. Example with a finite source: Equipment awaiting repair.

Models Single server, exponential service time (M/M/1) Single server, general service time (M/G/1) Multiple servers, exponential service time (M/M/s) Finite queue (M/M/s/K) Priority queue (nonpreemptive and preemptive) Finite calling population A Taxonomy — / — / — (and an optional fourth element / —) Arrival Service Number of Maximum Distribution Distribution Servers in Queue where M = Exponential (Markovian) D = deterministic (constant) G = general distribution Note that models 4 and 6 are not contained in the main text, but are included on the CD supplement for Chapter 14.

Notation Parameters: l = customer arrival rate m = service rate (1/m = average service time) s = number of servers Performance Measures Lq = average number of customers in the queue L = average number of customers in the system Wq = average waiting time in the queue W = average waiting time (including service) Pn = probability of having n customers in the system r = system utilization Arrival rate and service rate must be in the form of a rate (customers per unit time) and must be in the same units.