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**Introduction Queuing is the study of waiting lines, or queues.**

The objective of queuing analysis is to design systems that enable organizations to perform optimally according to some criterion. Possible Criteria Maximum Profits. Desired Service Level.

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Introduction Analyzing queuing systems requires a clear understanding of the appropriate service measurement. Possible service measurements Average time a customer spends in line. Average length of the waiting line. The probability that an arriving customer must wait for service.

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**Elements of the Queuing Process**

A queuing system consists of three basic components: Arrivals: Customers arrive according to some arrival pattern. Waiting in a queue: Arriving customers may have to wait in one or more queues for service. Service: Customers receive service and leave the system.

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**The Arrival Process There are two possible types of arrival processes**

Deterministic arrival process. Random arrival process. The random process is more common in businesses.

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The Arrival Process Under three conditions the arrivals can be modeled as a Poisson process Orderliness : one customer, at most, will arrive during any time interval. Stationarity : for a given time frame, the probability of arrivals within a certain time interval is the same for all time intervals of equal length. Independence : the arrival of one customer has no influence on the arrival of another.

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**The Poisson Arrival Process**

(lt)ke- lt k! P(X = k) = Where l = mean arrival rate per time unit. t = the length of the interval. e = (the base of the natural logarithm). k! = k (k -1) (k -2) (k -3) … (3) (2) (1).

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**HANK’s HARDWARE – Arrival Process**

Customers arrive at Hank’s Hardware according to a Poisson distribution. Between 8:00 and 9:00 A.M. an average of 6 customers arrive at the store. What is the probability that k customers will arrive between 8:00 and 8:30 in the morning (k = 0, 1, 2,…)?

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**HANK’s HARDWARE – An illustration of the Poisson distribution.**

Input to the Poisson distribution l = 6 customers per hour. t = 0.5 hour. lt = (6)(0.5) = 3. 1 2 3 4 5 6 7 8 k 1 2 3 (lt) e- lt k ! P(X = k )= 1 2 3 = 0! 1! 2! 3!

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**HANK’s HARDWARE – Using Excel for the Poisson probabilities**

Solution We can use the POISSON function in Excel to determine Poisson probabilities. Point probability: P(X = k) = ? Use Poisson(k, lt, FALSE) Example: P(X = 0; lt = 3) = POISSON(0, 1.5, FALSE) Cumulative probability: P(X£k) = ? Example: P(X£3; lt = 3) = Poisson(3, 1.5, TRUE)

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**HANK’s HARDWARE – Excel Poisson**

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**The Waiting Line Characteristics**

Factors that influence the modeling of queues Line configuration Jockeying Balking Priority Tandem Queues Homogeneity

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**Line Configuration A single service queue.**

Multiple service queue with single waiting line. Multiple service queue with multiple waiting lines. Tandem queue (multistage service system).

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Jockeying and Balking Jockeying occurs when customers switch lines once they perceived that another line is moving faster. Balking occurs if customers avoid joining the line when they perceive the line to be too long.

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**Priority Rules These rules select the next customer for service.**

There are several commonly used rules: First come first served (FCFS). Last come first served (LCFS). Estimated service time. Random selection of customers for service.

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**Tandem Queues These are multi-server systems.**

A customer needs to visit several service stations (usually in a distinct order) to complete the service process. Examples Patients in an emergency room. Passengers prepare for the next flight.

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Homogeneity A homogeneous customer population is one in which customers require essentially the same type of service. A non-homogeneous customer population is one in which customers can be categorized according to: Different arrival patterns Different service treatments.

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The Service Process In most business situations, service time varies widely among customers. When service time varies, it is treated as a random variable. The exponential probability distribution is used sometimes to model customer service time.

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**The Exponential Service Time Distribution**

f(t) = me-mt m = the average number of customers who can be served per time period. Therefore, 1/m = the mean service time. The probability that the service time X is less than some “t.” P(X £ t) = 1 - e-mt

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**Schematic illustration of the exponential distribution**

The probability that service is completed within t time units P(X £ t) = 1 - e-mt X = t

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**HANK’s HARDWARE – Service time**

Hank’s estimates the average service time to be 1/m = 4 minutes per customer. Service time follows an exponential distribution. What is the probability that it will take less than 3 minutes to serve the next customer?

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**Using Excel for the Exponential Probabilities**

We can use the EXPDIST function in Excel to determine exponential probabilities. Probability density: f(t) = ? Use EXPONDIST(t, m, FALSE) Cumulative probability: P(X£k) = ? Use EXPONDIST(t, m, TRUE)

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**HANK’s HARDWARE – Using Excel for the Exponential Probabilities**

The mean number of customers served per minute is ¼ = ¼(60) = 15 customers per hour. P(X < .05 hours) = 1 – e-(15)(.05) = ? From Excel we have: EXPONDIST(.05,15,TRUE) = .5276 3 minutes = .05 hours

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**HANK’s HARDWARE – Using Excel for the Exponential Probabilities**

=EXPONDIST(B4,B3,TRUE) =EXPONDIST(A10,$B$3,FALSE) Drag to B11:B26

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**The Exponential Distribution - Characteristics**

The memoryless property. No additional information about the time left for the completion of a service, is gained by recording the time elapsed since the service started. For Hank’s, the probability of completing a service within the next 3 minutes is ( ) independent of how long the customer has been served already. The Exponential and the Poisson distributions are related to one another. If customer arrivals follow a Poisson distribution with mean rate l, their interarrival times are exponentially distributed with mean time 1/l.

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**9.3 Performance Measures of Queuing System**

Performance can be measured by focusing on: Customers in queue. Customers in the system. Performance is measured for a system in steady state.

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**9.3 Performance Measures of Queuing System**

The transient period occurs at the initial time of operation. Initial transient behavior is not indicative of long run performance. n Roughly, this is a transient period… Time

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**9.3 Performance Measures of Queuing System**

The steady state period follows the transient period. n This is a steady state period……….. Roughly, this is a transient period… Meaningful long run performance measures can be calculated for the system when in steady state. Time

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**9.3 Performance Measures of Queuing System**

In order to achieve steady state, the effective arrival rate must be less than the sum of the effective service rates . k servers l< m For one server l< m1 +m2+…+mk For k servers with service rates mi l< km Each with service rate of m

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**Steady State Performance Measures**

P0 = Probability that there are no customers in the system. Pn = Probability that there are “n” customers in the system. L = Average number of customers in the system. Lq = Average number of customers in the queue. W = Average time a customer spends in the system. Wq = Average time a customer spends in the queue. Pw = Probability that an arriving customer must wait for service. r = Utilization rate for each server (the percentage of time that each server is busy).

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**For the case of an infinite population**

Little’s Formulas Little’s Formulas represent important relationships between L, Lq, W, and Wq. These formulas apply to systems that meet the following conditions: Single queue systems, Customers arrive at a finite arrival rate l, and The system operates under a steady state condition. L = l W Lq = l Wq L = Lq + l/m For the case of an infinite population

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**Classification of Queues**

Queuing system can be classified by: Arrival process. Service process. Number of servers. System size (infinite/finite waiting line). Population size. Notation M (Markovian) = Poisson arrivals or exponential service time. D (Deterministic) = Constant arrival rate or service time. G (General) = General probability for arrivals or service time. Example: M / M / 6 / 10 / 20

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**M/M/1 Queuing System - Assumptions**

Poisson arrival process. Exponential service time distribution. A single server. Potentially infinite queue. An infinite population.

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**M / M /1 Queue - Performance Measures**

P0 = 1 – (l/m) Pn = [1 – (l/m)](l/m)n L = l /(m – l) Lq = l2 /[m(m – l)] W = 1 /(m – l) Wq = l /[m(m – l)] Pw = l / m r = l / m The probability that a customer waits in the system more than “t” is P(X>t) = e-(m - l)t

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MARY’s SHOES Customers arrive at Mary’s Shoes every 12 minutes on the average, according to a Poisson process. Service time is exponentially distributed with an average of 8 minutes per customer. Management is interested in determining the performance measures for this service system.

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**MARY’s SHOES - Solution**

Input l = 1/12 customers per minute = 60/12 = 5 per hour. m = 1/ 8 customers per minute = 60/ 8 = 7.5 per hour. Performance Calculations Pw = l/m = r = l/m = P(X<10min) = 1 – e-2.5(10/60) = .565 m –l = 7.5 – 5 = 2.5 per hr. P0 = 1 - (l/m) = 1 - (5/7.5) = Pn = [1 - (l/m)](l/m)n = (0.3333)(0.6667)n L = l/(m - l) = 2 Lq = l2/[m(m - l)] = W = 1/(m - l) = 0.4 hours = 24 minutes Wq = l/[m(m - l)] = hours = 16 minutes

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**MARY’s SHOES - Spreadsheet solution**

=B4/B5 =1-B4/B5 =A11-B4/B5 =A11/B4 =B4/(B5-B4) =C11-1/B5 =H11*($B$4/$B$5) Drag to Cell AL11 =1-E11

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**Economic Analysis of Queuing Systems**

The performance measures previously developed are used next to determine a minimal cost queuing system. The procedure requires estimated costs such as: Hourly cost per server . Customer goodwill cost while waiting in line. Customer goodwill cost while being served.

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**Tandem Queuing Systems**

In a Tandem Queuing System a customer must visit several different servers before service is completed. Beverage Meats Examples All-You-Can-Eat restaurant

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**Tandem Queuing Systems**

In a Tandem Queuing System a customer must visit several different servers before service is completed. Meats Beverage Examples All-You-Can-Eat restaurant

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**Tandem Queuing Systems**

In a Tandem Queuing System a customer must visit several different servers before service is completed. Meats Beverage Examples All-You-Can-Eat restaurant A drive-in restaurant, where first you place your order, then pay and receive it in the next window. A multiple stage assembly line.

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**Tandem Queuing Systems**

For cases in which customers arrive according to a Poisson process and service time in each station is exponential, …. Total Average Time in the System = Sum of all average times at the individual stations

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**BIG BOYS SOUND, INC. Big Boys sells audio merchandise.**

The sale process is as follows: A customer places an order with a sales person. The customer goes to the cashier station to pay for the order. After paying, the customer is sent to the pickup desk to obtain the good.

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**BIG BOYS SOUND, INC. Data for a regular Saturday Personnel.**

8 sales persons are on the job. 3 cashiers. 2 workers in the merchandise pickup area. Average service times. Average time a sales person waits on a customer is 10 minutes. Average time required for the payment process is 3 minutes. Average time in the pickup area is 2 minutes. Distributions. Exponential service time at all the service stations. Poisson arrival with a rate of 40 customers an hour.

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BIG BOYS SOUND, INC. Only 75% of the arriving customers make a purchase! What is the average amount of time, a customer who makes a purchase spends in the store?

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**BIG BOYS SOUND, INC. – Solution**

This is a Three Station Tandem Queuing System Pickup desk M / M / 2 (.75)(40)=30 Cashiers M / M / 3 l = 30 Sales Clerks M / M / 8 l = 30 W3 = 2.67 minutes l = 40 W2 = 3.47 minutes Total = minutes. W1 = 14 minutes

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