Queuing Theory Queuing theory deals with the analysis and management of waiting lines. Service Facilities Fast-food restaurants, Post office, Grocery store, Bank Theme parks - Disneyland Highway traffic Manufacturing Equipment awaiting repair Phone or computer network Product orders Queuing models are used to: describe the behavior of queuing systems determine the level of service to provide evaluate alternate configurations for providing service

C. Westin’s Barber Shop Charlotte Westin is a popular barber who runs a one-woman barber shop. Her family owns the Westin hotels. Ms. Westin opens her shop at 8:00 A.M. and fills up quickly! The table shows her queuing system in action over a typical morning. Customer Time of Arrival Haircut 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 —

Number of Customers in the System

Modeling queuing systems Major inputs: Description of the system Arrival and Service patterns (rates and distributions) Number of servers Queue discipline and capacity Population size – infinite or finite Analytical approach Quick analyses are possible as long as “already invented” Provide steady state results for various performance measures For complex or “simple” systems that are somewhat different than the existing analytical models, it is very difficult to develop new models, therefore Simulation approach Tends to be the only viable alternative for complex queuing systems Can provide a wide range of results and allow experimentation

Arrival Pattern The time between consecutive arrivals to a queuing system are called the interarrival times Ex: Barber shop interarrival times 12, 10, 5, 35, and 38 minutes The expected number of arrivals per unit time is referred to as the mean arrival rate usually denoted by l (lambda) The mean of the probability distribution of interarrival times is 1/l = Expected interarrival time Most queuing models assume that the form of the probability distribution of interarrival times is an exponential distribution.

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 queuing 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. Exponential interarrival times imply Poisson distribution for the number of arrivals in a given period.

Service Pattern When a customer enters service, the elapsed time from the beginning to the end of the service is referred to as the service time The expected number of service completions per unit time is referred to as the mean service rate usually denoted by m (mu) The symbol used for the mean of the service time distribution is 1/m = Expected service time Either an exponential distribution is assumed Decent approximation if the jobs to be done are random. Not a good approximation if the jobs to be done are always the same (e.g., fixed set of tasks) Or any (general) distribution Only single-server model is easily solved.

Queue and Population Source 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. Most queuing 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.

Commercial Service Examples System type Customers Server(s) Bank teller services People Teller ATM machines services People ATM machine Store checkout People Clerk Window repair services Windows Repair person Airport check-in counters People Airline agent Call centers for catalog sales People Phone agent Call centers for tech help People Technical rep Gas station Cars Pumps Dental services People Dentist

Internal Service Examples System type Customers Server(s) Secretarial services Employees Secretary Copying services Employees Copy machine Production systems Jobs Machine (center) Maintenance systems Machines Repair crew Inspections station Items Inspector Tool crib Operators Clerk Semiautomatic machines Machines Operator X-ray services Doctors X-ray technician

Transportation Service System Examples System type Customers Server(s) Ambulance service (EMS) People Ambulance Fire department Fires Fire truck Truck (un)loading dock Trucks Crew Port (un)loading dock Ships Crew Air traffic Airplanes Runway Expressway tollbooth Vehicles Cashier Parking deck Vehicles Parking space Elevator service People Elevator

System Performance Measures The most common measures are: Average number of customers waiting in the system (denoted by L ) or Average time customers wait in the system (denoted by W ) Average number of customers in the queue (denoted by Lq) or Average time customers wait in the queue (denoted by Wq) The above metrics assume that the system is in a steady-state condition. 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.

More on Performance Measures 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 queuing 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 or P(W > 2 hours) < 0.05

Analytical approaches for basic systems Kendall notation: __ /__ /__ (and an optional fourth element) arrival process/service process/no. of servers /Q capacity Where M = Exponential (Markovian) and G = general distribution Models available in the text and also on Q.XLS 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) Finite calling population

The Single Server Model (M/M/1) 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 Fundamental formulas: The utilization factor r = l / m P0 = 1 - l / m = 1 - r and Pi = (1 – r ) r i L = r / (1 – r) = l / (m – l) W = (1 / l)L = 1 / (m – l) Wq = W – 1/m = l / [m(m – l)] Lq = lWq = l2 / [m(m – l)] = r2 / (1 – r)

The M/M/1 Example Customers arrive to a small-town post office at an average rate of 10 per hour (Poisson distribution). There is only one postal employee on duty and he can serve customers in an average of 5 minutes (exponential distribution). Develop a spreadsheet to compute various performance measures.

M/G/1 Model and Example Assumptions: P0 = 1 – r 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). P0 = 1 – r Lq = [l2s2 + r2] / [2(1 – r)] L = Lq + r Wq = Lq / l W = Wq + 1/m

An Example: ABC Car Wash is an automated car wash. Each customer deposits four quarters in a coin slot, drives the car into the auto-washer, and waits while the car is automatically washed. Cars arrive at an average rate of 20 cars per hour (Poisson). The service time is exactly 2 minutes. Compute various performance measures (spreadsheet).

M/M/s Model and Example 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 becomes r = l / sm The rest of the formulas are too messy to show here. An Example: A convenience store has three registers open. Customers arrive to check out at an average of 1 per minute (Poisson). The service time averages 2 minutes (exponential). Compute various performance measures (spreadsheet).

Economic Analysis of Queuing Systems Typically manager is interested in minimizing the total cost. 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 to Minimize TC = SC + WC 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 or WC = Cw Lq No. of servers Expected cost per unit time total cost service cost waiting cost

Example 1 The MIS department of a high tech company handles employee requests for assistance when computer questions arise. Employees requiring assistance phone the MIS department with their questions (but may have to wait on hold if all of the tech support staff are busy). The MIS department receives an average of 40 requests for assistance per hour (Poisson). The average question can be answered in 3 minutes (exponential). The MIS staff is paid an average of $15 per hour. The average employee earns $25 per hour. Question: What is the optimal size of the MIS tech support staff? Via Spreadsheet

Example 2 “Miller Manufacturing…” Miller Manufacturing owns 10 identical machines used in the production of colored nylon thread for the textile industry. Machine breakdowns occur following a Poisson distribution with an average of 0.01 breakdowns occurring per operating hour per machine. The company loses $100 each hour a machine is inoperable. The company employs one technician to fix these machines when they break down. Service times to repair the machines are exponentially distributed with an average of 8 hours per repair. (So service is performed at a rate of 1/8 machines per hour.) Management wants to analyze the impact of adding another service technician on the average length of time required to fix a machine. Service technicians are paid $20 per hour Via Spreadsheet

Example 3 “Zippy-Lube…” Zippy-Lube is a drive-through automotive oil change business that operates 10 hours a day, 6 days a week. The profit margin on an oil change at Zippy-Lube is $15. Cars arrive randomly at the Zippy-Lube oil change center following a Poisson distribution at an average rate of 3.5 cars per hour. The average service time per car is 15 minutes (or 0.25 hours) with a standard deviation of 2 minutes (or 0.0333 hours). A new automated oil dispensing device cane be acquired for $5,000. The manufacturer's representative claims this device will reduce the average service time by 3 minutes per car. (Currently, employees manually open and pour individual cans of oil.) The owner would like to analyze the impact the new automated device would have on his business and determine the pay back period for this device. Via Spreadsheet

Example 4 A McDonalds franchise is trying to decide how many registers to have open during their busiest time, the lunch hour. Customers arrive during the lunch hour at a rate of 98 customers per hour (Poisson distribution). Each service takes an average of 3 minutes (exponential distribution). Question #1: If management would not like the average customer to wait longer than five minutes in line, how many registers should they open? Question #2: If management would like no more than 5% of customers to wait more than 5 minutes, how many registers should they open? Via Spreadsheet

Land’s Beginning Calls arrive at a rate of 150 per hour to the 800 number for the Lands Beginning mail-order catalog company. The company currently employs 20 operators who are paid $10 per hour in wages and benefits and can each handle an average of six calls per hour. Assume that inter-arrival times and service times follow the exponential distribution. A maximum of 20 calls can be placed on hold when all the operators are busy. The company estimates that it costs $ 25 in lost sales whenever a customer calls and receives a busy signal. On average, how many customers are waiting on hold at any point in time? What is the probability that a customer will receive a busy signal? If the number of operators plus the number of calls placed on hold cannot exceed 40, how many operators should the company employ? If the company implements your answer to part c, on average, how many customers will be waiting on hold at any point in time and what is the probability that a customer will receive a busy signal?

Some Insights About Designing Queuing Systems 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. Decreasing the variability of service times (without any change in the mean) improves the performance of a queuing system substantially. 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. Applying priorities when selecting customers to begin service can greatly improve the measures of performance for high-priority customers.

Psychology of waiting Unoccupied time feels longer than occupied time Uncertainty makes waits seem longer Unfair waits are longer than fair waits Unexplained waits are longer than explained waits The more valuable the service, the longer the customer is willing to wait

LL Bean – Catalog sales LL Bean’s mail order business Mail order phone lines open 24 hours per day, 365 days per year 78,000 calls per week (average) Seasonal variations as well as variability during each day How LL Bean estimates the number of servers needed Each of the week’s 168 hours in a week is modeled separately as a period to be staffed Each hour modeled as an M/M/s queue Arrival rates and service rates estimated from historical data Service standard: no more than 15% of calls wait more than 20 seconds Full-time, part-time, and temporary workers scheduled to meet service standard

Practice Problems and Cases Suggested practice problems: 6, 7, 8, 9, 14, 17, 20, 23, 26 Group exercises: Case 13.3