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LESSONs NINE and TEN QUEUING MODELS

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**Queuing Models 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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**Elements of the queuing process**

A queuing system consists of five basic components: Calling population Arrival process Queue configuration Queue discipline Service process

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CALLING POPULATION

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CALLING POPULATION Homogeneous or heterogeneous Can be finite by limiting calls or infinite if population is big

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ARRIVAL PROCESS

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ARRIVAL PROCESS Static no control Dynamic control by machines or man Balking not joining the queuing Reneging leave line before service

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QUEUE CONFIGURATION

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QUEUE CONFIGURATION Refers to number of queues, locations, spatial requirement and effect on customer behavior Jockeying – line switching activity

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QUEUE DISCIPLINE

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SERVICE PROCESS

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**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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**The Arrival Process The random process is more common in businesses.**

Under three conditions a Poisson Distribution can describe the random arrival process.

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**The three conditions required for the existence of the Poisson arrival 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. These conditions are unrestrictive and are approximately satisfied in many situations.

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**The Poisson Arrival Distribution**

X k e ! = l t) t ( ) - 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 (An illustration of the Poisson distribution)**

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

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**SOLUTION 1 2 3 Input to the Poisson distribution**

SOLUTION Input to the Poisson distribution l= 6 customers per hour. t = 0.5 hour. l t = (6)(0.5) = 3. 1 2 3 4 5 6 7 8 3 2 1 1 = 2 3 0! 3! 2! 1!

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**Measures of Queuing System Performance**

P0 = Probability that there are no customers in the system. Pn = Probability that there are “n” customers in the 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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**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 Characteristics Poisson arrival process.**

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

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**Performance Measures for the M / M /1 Queue**

P0 = 1- (l / m) Pn = [1 - (l / m)] (l/ m)n L = l / (m - l) Lq = l 2 / [m(m - l)] W = 1 / (m - l) Wq = l / [m(m - l)] Pw = l / m r = l / m

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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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**SOLUTION P0 = 1- (l / m) = 1 - (5 / 7.5) = 0.3333**

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 P0 = 1- (l / m) = 1 - (5 / 7.5) = Pn = [1 - (l / m)] (l/ m) = (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 Pw = l / m = r = l / m =

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m l WINQSB Input Screen

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**Performance Measurements**

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**M / M / k Queuing Systems Characteristics**

Customers arrive according to a Poisson process at a mean rate l. Service time follow an exponential distribution. There are k servers, each of which works at a rate of m customers. Infinite population, and possibly infinite line.

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Performance Measure

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**The performance measurements L, Lq, Wq,, can be obtained**

from Little’s formulas.

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Revision Question At a Food Lion Provision Shop, customers spend an average of 25 mins selecting their groceries and checking out by entering a single line queue served by two cashiers. The service times required for the cashiers to check out customers follow an exponential distribution and average four minutes. Customers arrive at the cashier counter according to a Poisson distribution at the average rate of eight customers per hour The table below shows part of the computer calculation: Determine the followings : average time a customer spends in the store (in minutes). average number of customers waiting in line prior to being checked out. proportion of customers who will have to wait in line. What assumption(s) did you make in part (a)? Comment whether the assumption(s) is/are realistic.

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**Queuing Process Multiple queue advantages :**

Service can be differentiated Division of labour possible Selection option for customer Deterred balking

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**Types of services clearly stated**

Queuing Process Multiple queue disadvantages : Anxiety Lack of fairness Lack of privacy Types of services clearly stated

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**Queuing Process Single queue advantages :**

First come first serve (fairness) No anxiety to select fastest line Reneging difficult Queue cutting resolved Privacy enhanced Reducing average waiting time

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**Unnecessary held up in waiting line**

Queuing Process Single queue disadvantages : No specialisation Unnecessary held up in waiting line Possible balking

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**Free to wander about and browse items**

Queuing Process Take a number advantages : No need for formal line Free to wander about and browse items More relaxing

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**Require large waiting area**

Queuing Process Take a number disadvantages : Reneging Require large waiting area

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**Queuing Process QUEUE DISCIPLINE**

Policy of selecting next customer from the queue for service First come first serve most popular static method (selection depends on position in queue only)

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Queuing Process Dynamic disciplines selection involve some attribute or status in selection Shortest processing time minimise average time of customers

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Queuing Process Preemptive priority interrupt current person service for newly arrived customer with higher priority e.g. fire or ambulance services Round robin service give customer partial service and then movers to next waiting customer – alternating between waiting and being served

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Queuing Process Service person begins to take orders while customers are still waiting in line is a direct approach to avoid reneging

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**Queuing Process SERVICE PROCESS**

Distribution of service times, arrangement of servers, management policies and server behaviour contribute to performance

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**WAITING PERCEPTION Unoccupied time feels longer than occupied time**

Preprocess waits feel longer than in process waits Anxiety makes waits seem longer

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**WAITING PERCEPTION Uncertain waits are longer than known finite waits**

Unexplained waits are longer than explained wait Unfair waits are longer than equitable waits

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**WAITING PERCEPTION People more willing to wait for valuable service**

Solo waiting feels longer than group waiting Customer attitudes

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**WAITING PERCEPTION Environment**

Unused facilities and idle staff increase annoyance Unfamiliar music makes perceived time seem longer than familiar background music

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**REVIEW QUESTIONS 1. What does M/D/1 refers to?**

2. What are the characteristics of M / G / 1? 3. How can you measure the performance of a queue?

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REVIEW QUESTIONS 4. List and discuss the factors affecting queue perception. 5. What are the five features of a queue? 6. How can you configure queue?

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1 Elements of Queuing Theory The queuing model –Core components; –Notation; –Parameters and performance measures –Characteristics; Markov Process –Discrete-time.

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