OM0010-SLM-Unit-11

Operations Management

Unit 11

Unit 11

Queuing Problems

Structure 11.1

Introduction Objectives

11.2

Meaning of Queuing Problems Applications of queuing models Elements of the queuing model Some important notations

11.3

Pure Birth Model

11.4

Pure Death Model

11.5

General Poisson Model

11.6

Specialised Poisson Queues

11.7

Single Server Model: (M/M/1):( /FCFS)

11.8

Multiple Server Model: (M/M/S): (M/M/S): (/FCFS)

11.9

Self Service Model: (M/M/∞): (GD/∞/∞)

11.10 Machine Serving Model: (M/M/R) R
11.1 Introduction The previous unit discussed about the inventory management in detail. Apart from maintaining inventory, it is also essential to quickly supply the inventory to the required customers so that they do not have to wait for a

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long time. Queues are generally formed when a number of customers customers wait to receive a particular product or service. Queues are seen at banks, restaurants, medical clinics, railway stations, and bus stops. Queues help manage customers and service delivery. However, queues may also result in problems. A problem that involves waiting for a service is called a queuing problem or a waiting line problem. Long queues frustrate customers and may become a reason for a customer to look for other options. Therefore, organisations need to solve queuing problems to retain customers, as well a s to attract new customers. A queuing problem can be solved by b y using the queuing theory, which is a mathematical study of waiting lines. The queuing theory aims to maintain a balance between the waiting time of customers and the idle time of the service provider. In this unit, you will study the concept of queuing. You will also study about queuing problems and the applications and elements of queuing models. In addition, you will study about the importance of queue management along with the different models of queuing. Objectives: After studying this unit, you will be able to: 

define the concept of queuing



elaborate on queuing problems



identify the applications of queuing models



explain the importance of queue management



discuss the different models of queuing

11.2 Meaning of Queuing Problems Queues are generally formed when customers requiring service wait for getting the service. Queues or waiting lines are generally seen at banks, railway ticket counters, temples, clinics, petrol pumps, and bus stops.


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Figure 11.1 is a depiction of a queue system wherein there are three servers or counters and there are three clients:

Figure 11.1: Depiction of a Queue System

The clients are making the request to the server to get their issues resolved. After the issues are resolved, the client leaves the server to enable other persons to get their request catered to. Queues are formed when customers (human beings or physical entities), who require service, wait either due to the lack of service facilities or when service facilities do not work efficiently and take more time than required. The queuing theory can be applied to situations where it is not feasible to anticipate the arrival rate of customers and the rate of service provided by the service provider. It is used to determine the level of service that balances the following two costs: Cost of offering the service: service: Involves the cost that is associated with service facilities and their operations Cost incurred due to delay in offering service: Involves the cost associated with customers’ waiting time Apart from this, the different performance measures for analysing the queuing theories are as follows: The distribution distribution of the waiting time and the staying time of a customer: The staying time is the waiting time plus the service time. If there are short queues, it either means good customer service or too much capacity. If there are long queues, it means low capacity or bad customer service.


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The distribution of the number of customers in the system: The rate of customer arrival can be affected by the management through strategies such as advertising or differential pricing. The distribution of the amount of work in the system: This is given as: Distribution of the amount of work = sum of service times of the waiting customers + residual service time of the cu stomer in service The distribution of the busy period of the server: This is a period of time during which the server works continuously. The long line does not always reflect long waiting lines if the service rate is fast. They may indicate a need for adjusting the service rate of the system or changing the arrival rate of customers. NOTE: The queuing theory is also known as the waiting line theory. The queuing theory was propounded by A. K. Erlang, a Danish engineer, in 1903. He recognised the problem of congestion of telephone traffic and suggested how to minimise the delay time in between calls. However, over a passage of time, the queuing theory is being used to determine the average number of customers waiting in line and the average waiting time of the customers. The theory provides a statistical description of queues’ behaviour. Queuing problems are solved using the queuing models, which are based on the assumptions of the queuing theory. Let us study the different applications of the queuing models. 11.2.1 Applications of queuing models Customers are the primary source of revenue for an organisation. They are satisfied if the organisation provides products or services at minimum cost and within the stipulated time. If an organisation makes unwanted delays in delivering services, customers may become highly dissatisfied and switch to other brands. Therefore, the waiting time of customers and the cost of providing services should be minimised. This can be done by using queuing models. The applications of queuing models are:


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Industrial manufacturing or production process: It is a field that has a wide application of queuing models. In manufacturing or production processes, queuing models are used to minimise the time of the following: 

Assembly lines



Tool room service



Billing



Computer centres



Expensive stock items

Transportation: In this field, queuing models are used to minimise the waiting time of the arrival and departure of buses, and checking and ticketing counters. Communication: In the communication industry, queuing models are applied to minimise the waiting time between the telephone calls that need to be attended by agents. Service industry: In this field, queuing models are used to minimise the customer waiting time. Human resource management: In this field, queuing models are used to determine appropriate waiting time for the promotion of an employee. 11.2.2 Elements of the queuing model Having studied the basic concepts of the applications of the queuing model, let us now discuss its basic elements. The basic elements of the queuing model are: Input source or arrival pattern: This indicates a pattern in which customers arrive to receive a service and join queues. Since customers arrive in a random manner, their arrival pattern is represented in the form of probabilities. The number of customers and their arrival time distribution characterises these patterns. Queuing process: It is the formation of the number of queues (customer is waiting for service) and their respective lengths. The number of queues depends on the design of the service system. The queuing system may have a single service channel (for example, a doctor’s clinic) and customers Sikkim Manipal University

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served per unit of time. Such a system is called the server model or the single channel model. On the other hand, if the system has two or more service channels (for example, banks), it is called multiple server model. Queuing disciplines: It is a manner in which customers waiting in the queue are served. The most common discipline is First Come First Served (FCFS). However, there are a number of ways in which customers in the queue are served. Some of them are: 

Static queue discipline: Includes FCFS and Last Come First Served (LCFS).



Dynamic queue discipline: Includes Service in Random Order (SIRO), priority class service, pre-emptive priority, and non-pre-emptive priority.

Service mechanism (or service pattern or service process) facilities: These facilities are characterised by their arrangement and service time distribution. Service mechanism can be determined when the number of customers that can be served at a time, is known. It is also necessary to determine the statistical distribution of service time and the time when the service will be available. In most situations, service time is a random variable with the same distribution for all arrivals, but there are cases where there are clearly two or more classes of customers. The service mechanism is characterised by the capacity of service facilities, distribution of the service time, server’s behaviour, and management policies. Customer behaviour: It plays a vital role in the study of queues. In general, customer behaviour is influenced by the length of the queue. For example, a customer may leave if the queue is too long (balking of queue) or he/she does not have time to wait (reneging of queue) or there is not sufficient waiting space (jockeying of queue). However, in some situations, customers may move from one waiting line to another due to urgency. Statistical tools: These tools help in the estimation of the actual arrival and service time distributions of customers. Some of the tools are the Poisson distribution and the exponential method. These tools help in describing the expected value of various operational characteristics of the queuing process.


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11.2.3 Some important notations Some of the important notations that are widely used in the application and management of queues are given in Table 11.1: Table 11.1: Some Important Notations used in the Queue System

λ Μ 1/λ 1/μ ρ

Average utilisation of

Ls

Average number of customers (or units) in the system

Ws

Average time a customer spends in the system

Lp

Average number of customers in the waiting line or queue

Wq

Average time a customer spends waiting in the queue

n

Number of arrivals per certain time period Number of customers (units) served per certain time period Mean time between arrivals Mean time per customer served

the service facility (λ/μ)

Number of customers in the service system

Self Assessment Questions: 1.

What are queues?

2.

The queuing theory was propounded by A. K. Erlang. (True/false)

3.

Which of the following options is not an element of a queuing model? a. Output source b. Queuing process c. Queuing disciplines d. Service mechanism

4.

In which of the following areas is the queuing model not applicable? a. Transportation b. Communication c. Research d. Service industry


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11.3 Pure Birth Model In the pure birth model, customers are determined to get service at any cost. They join a queue with the intention of never leaving the queue, no matter how much time it will take to get their service request processed. The pure birth model is used for estimating the arrival time distribution of customers. This model is based on an assumption that after customers arrive at the queuing system, they never leave the queue without their request being processed. The main objective of this model is to derive an expression for the probability Pn (t) of n arrivals during time interval (t+ Δt). The following are some of the important terms that are used in the development of queuing models:

Δt = Time interval that is so small that the probability of the arrival of more than one customer is negligible. This implies that during any given small

interval of time (Δt), only one customer can arrive. λ Δt = Probability that a customer will arrive at the system during time (Δt). 1-λ

Δt = Probability that no customer will a rrive (Δt).

in the system during time

If the arrivals are completely random, then the probability distribution of a number of arrivals in a fixed time interval follows the Poisson distribution. There are two conditions for applying the Poisson distribution, which are as follows: Distribution of inter-arrival time (exponential distribution): Refers to the condition in which the number of arrivals (n) has joined the queue in time t. In such a case, the Poisson distribution will be:

P n (t )  ( t ) n / n!e   t n = 0, 1, 2, 3….. The exponential distribution would be applied if an associated random variable is defined as the inter–arrival time t. The mathematical expression of exponential distribution is as follows:

f (t )   e  t Markovian property of inter-arrival time: Refers to the condition of the probability that the service provided to a customer will be completed at Sikkim Manipal University

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some time t. This time t is independent of how long the customer has received the service. The mathematical expression of the Markovian property of inter-arrival time is as fo llows: Prob [T ≥ t1│T ≥ t0] = Prob [0 successive arrivals.

≤ T ≤ t 1-

t0] Where T is the time between

Let us discuss the pure birth model with the help of an example. Example 1: In India, birth rate is 1 birth every 15 minutes and the time between births follows the exponential distribution. In such a case, find the following: 

The average birth rate in India per year



The probability of no births in any one day

Solution: a.

The birth rate per day is λ = (24x60)/15 = 96 births per day

The average number of births per year = λt = 96 x 365 = 35040 births per year

b.

The probability of no births in any day computed from the Poisson distribution is as follows: P0(1) =[(96x1)0]/ 0! e--96x1 = 0


What is the use of the pure birth model?

6.

______________refers to the condition of the probability that the service provided to a customer would be completed at some time t.

7.

Why is the pure birth model used? a. For estimating the arrival time distribution of customers b. For estimating the waiting time distribution of customers c. For estimating the long time distribution of customers d. For estimating birth per day


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11.4 Pure Death Model In the pure death model, it is assumed that none of the customers have joined the system and the service is continued for those who are already in the system. In this model, no new person is allowed to join the queue. The service is available only for those who are already in the queue. This kind of model is widely observed in day to day operations. For example, when lunch or closing time approaches in a bank, the bank manager does not allow any new person to join the queue, while he/she continues to provide service to the people who are already in the queue. This model is known as the pure death model.

≥ 1 number of customers in the system. As the service rate is μ, the rate at which the customers are leaving the system after servicing is also μ. The process of Let us assume that at a starting time (t = 0) there are N

customers leaving the system is called the pure death process. The pure death process is also termed ‘distribution of departures’. The distribution of departures from the system takes place on the basis of the following three axioms: 

Probability of the departure of one customer in time Δt is μ Δt



Probability of departure of more than one customer between time t and t

+ Δt is negligible 

The number of departure in non-overlapping intervals are statistically independent

Some of the notations used in the pure death model are as follows:

μΔt = Probability that a customer in service at tim e service during time Δt

t will receive complete

μΔt = Probability that the customer in service at time t will not receive complete service during time Δt 1-

P n (t ) 

(  t ) N n e   t ( N  n)!


for n = 1, 2, 3 …… N

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P n (t )  1 

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 N n1 (  t ) N n e   t (N - n)!

for n = 0

Note: The number of departures in time t follows ‘Truncated Poisson Distribution’. Let us study the pure death model with the help of an example. Example 2: A fruit seller stores a stock of 18 dozen mangoes at the beginning of every week. On an average, the fruit seller sells 3 dozens per day (one dozen at a time), but the actual demand follows a Poisson distribution. Whenever the stock level reaches 5 dozens, a new order of 18 dozens is placed for delivery at the beginning of the following week. Determine the probability of placing an order in any one day of a week. Solution:

Purchases occur at the rate of μ = 3 dozens per day, thus, the probability of placing an order by the end of day t is given as follows:

P n 5 (t ) 

(3t )18n e 3t (18  n)!

for n = 1, 2, 3 ……18 and t  1,2,3,4,...7

Pn≤5 (t) = P0 (t) + P1 (t) +………P5 (t) The calculations of Pn≤5 (t) is as follows: t (days)

1

2

3

4

5

6

7

μt

3

6

9

12

15

18

21

Pn≤5 (t)

0

.0088

.1242

.4240

.7324

.9083

9755


In ____________, it is assumed that none of the customers have joined the system and the service is continued for those who are already in the system.

9.

The process of customers leaving the system is called the pure death process. (true/false)


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11.5 General Poisson Model While dealing with a queue, the number of people arriving or leaving the queue is not fixed. This means that the rate of arrival or departure, or the number of people jumping the queue is a random process. In the theory of probability, the Poisson process deals with random process or processes, which are involved in the counting of number of such events with respect to the time period. The time period between each consecutive event follows an exponential curve with a parameter λ. The general Poisson model combines both, arrivals and departures, based on the assumption that inter-arrival time and service distribution time follow exponential distribution. The development of this model is based on the long-run or steady-state behaviour of the queuing situation. This type of behaviour is achieved after the system has been in operation for a sufficiently long time. In the general Poisson model, it is assumed that the arrival and departure rates depend on the number of customers in the service facility. Some of the notations used in the general Poisson model are as follows: n = Number of customers in the system

λ = Arrival rate μ = Departure rate Pn = Steady - State probability of n customers in the system In this model, Pn is the function of λ and μ. These probabilities are used to determine the system’s measurement of performance, such as the average queue length, the average waiting time, and the average utilisation of the facility. The probabilities Pn are determined by using the transition rate. The queuing system is in state n when the number of customers in the system is n. The value of n can be changed only in the following two possible states: 

n-1 when a departure occurs at μ rate



n+ 1 when an arrival occurs at λ r ate


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For n>0, the expected rates of flow into and out of state n must be equal. Based on the fact that state n can be changed to states n-1 and n+1 only, we get the following:

Expected rate of flow into state n = λ n-1Pn-1+μn+1 Pn+1 Similarly, expected rate of flow out of state n = (λ n + μn) Pn Equating the two rates, we get the following balance equations:

λn-1 P n-1 + μn+1 Pn+1 = (λn+μn)Pn, n = 1, 2, 3, The balance equation associated with n = 0 is λ 0P0 = μ1P1 The balance equations are solved recursively in terms of P0 is as follows, for n=0 P1 = (λ0/μ1) P0 For n = 1

λ0P0 +μ2P2 = (λ1+μ1)P1 Substituting P1 = (λ0/μ1) P0, we get: P2 = (λ1λ0/μ2μ1) P0 In general, the value of Pn is calculated as follows: Pn = (λn-1λn-2…… λ0/μnμn-1………μ1) P0, n = 1. 2. .. The value of P0 is determined from the equation as follows:

 n0 P n  1 11.6 Specialised Poisson Queues Until now we have discussed the concept of single server, single queue using the Poisson distribution. Let us now discuss the case of specialised Poisson queues. A waiting customer is selected from the queue to start

service with the first available server. The arrival rate at the system is λ customers per unit time. The service rate for any server is μ customers per unit time. The total number of customers in the system includes the customers receiving the service and those who are waiting in the queue. Figure 11.2 shows the specialised queuing situation with parallel servers:


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Figure 11.2: Queuing System

Figure 11.2 shows a waiting customer deciding to move or select any one of the server, which is available for processing his request. The standard notation for representing the arrivals and departures distribution are as follows: M = Markovian (or Poisson) arrival or departure distribution D = Constant time Ek = Erlang or gamma distribution of time (or the sum of independent exponential distribution) G = General distribution of service time The queue discipline notations are as follows: FCFS = First come, first served LCFS = Last come, first served SIRO = Service in random order GD = General discipline The model (M/D/20): (GD/25/∞) refers to Poisson arrivals (or exponential inter-arrivals time), constant service time and 20 parallel servers. The queue discipline is GD, and there is a limit of 25 customers on the entire system. The size of the source f rom which customers arrive is infinite. The most commonly used measures of performance in a queuing situation are called steady-state measure of pe rformance, which are as follows: Sikkim Manipal University

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Ls = Expected number of customers in a system Lq = Expected number of customers in a queue Ws = Expected waiting time in a system Wq = Expected waiting time in a queue s = Expected number of busy servers Self Assessment Questions: 10. The general Poisson model combines both arrivals and departures based on an assumption that inter-arrival time and service distribution time follows exponential distribution. (true/false) 11. FCFS stands for ________________. 11.7 Single Server Model: (M/M/1):( /FCFS) The single server model helps in solving the queuing problems with a single channel to serve customers. It is one of the most widely used and simplest queuing models. This model is based on the following assumptions about the queuing system: 

Inter-arrival time follows exponential distribution and arrival time follows Poisson distribution



Single waiting line with infinite capacity



Service procedure is FCFS



Single server with exponential distribution of service time



Every arrival waits to be served regardless of the length of the line (that is no balking or reneging)



The average service rate is greater than the average arrival rate



Service time also differs from one customer to another and is independent of one another

There are two types of single server models namely single server single queue model and single server multiple queue model. Sikkim Manipal University

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Single server single queue In single server single queue, the models involve one queue and one service counter. The customer waits until the service is provided to him/her and after that the customer departs. Figure 11.3 depicts the diagrammatic representation of the single - server single queue:

Figure 11.3: Single Server Single Queue

Single server several/multiple queues In the case of single server several queues, there is more than one queue. Figure 11.4 represents the diagrammatic version of single server several queues:

Figure 11.4: Depicting Single Server Several Queues

In the figure.11.4 we observe that there is a single server, in the form of an ATM machine and there are several queues with the lady standing in a new queue. Figure 11.5 depicts another version of a queue wherein it is planned for multiple servers, but due to some constraints, it is acting like a single


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server. The constraints can be technical default or unavailability of the service provider.

Figure 11.5: Depicting Single Server, Multiple Queues

In the case of single server multiple queues, the job of the queue manager becomes more challenging because he/she has to implement control, before being in a position to serve the customers. If the queue system is in state n (number of customers) at time t, the following events may occur during a small interval of time: 

The system contains ‘n’ number of customers and there is no arrival and departure of customers



The system contains ‘n+1’ number of customers and there is no arrival and departure of customers



The system contains ‘n-1’ number of customers and there is no arrival and departure of customers

The following are the quantitative expressions that are used for analysing single-channel waiting lines:

λ = Mean or expected number of arrivals per time period μ = Mean or expected number of items served per time period ρ = λ/μ =


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The following are operating characteristics of a single-channel waiting line: The probability of number of units in the system P0 = 1 –

(λ/μ)

Probability of ‘n’ number of units in the system (units in queue + number of units being served) Pn = (λ/μ)n, When

 n 0 P n  1

Average number of units in the system

E ( n)orLS   n0n. P n 

 P0 [1  (  )]2 

  n0 np0 (   )n

   (1 -  / ) [1  (  )]2 



  - 

Mean (expected or average) number of units in the queue waiting for service: E(m) or Lq = Expected number in the system – Expected number in service



   



 



 2  (    )

Mean (expected) waiting time in system (time in queue plus service time):

Ws 

Expected number in the system Expected rate of arrival





   



1   

Mean (expected or average) time a unit spends waiting in the queue:

Ws 

Expected number in the queue Expected rate of arrival

 2



 (    ) 

  /  (

1

)   

Probability that the queue size is greater than or equal to k:


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P ( n  k )   n 0 P n  ( P 0  P 1  ....  P k 1 )

 1  { P 0   /  P 0  .....  ( /  ) k 1 P 0 } P 0{1  ( /  ) k } 1  ( /  ) k {1  ( /  )} The number in the system should be at least 2 for a non-empty queue (one in queue and one under service). The probability of a non-empty queue:

  n0 P n  ( P 0  P 1 )  1{ P 0  ( /  ) P 0 Average length of a non-empty queue =



Average length of a queue Probabilit y of a non - empty queue



 2  (    )

( /  ) 2



   

Probability that the waiting time in a queue is greater than and equal to t is: P (waiting time

≥ t) 

 

e (    )t

Let us study the single server model with the help of an example. Example 3: PQR is a warehouse with one loading port that is handled by a three-person crew. The arrival rate of trucks at the loading port is two trucks per hour, which is Poisson distributed. The average time for loading a truck is 150 minutes, which is exponentially distributed. The operating cost of a truck is ` 13 per hour and the crew members get ` 4 per hour for loading a truck. Should the truck owner add one more crew for loading a truck? Solution:

Given that the average arrival time λ = 4 per hour Loading truck rate μ = 300/60 = 5 per hour for the existing crew (three members) Number of loaders = 3 Ws = 1/ (μ - λ) = 1/ (5-4) = 1 Total hourly cost = Loading crew cost + cost of waiting time


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= {(Number of loaders) x (Hourly wage rate)} + {(Expected waiting time per truck, W s) (Expected arrival per hour, λ) x (Hourly waiting cost)} = {3 x 4} + {(1/1) x 2 x 13} = ` 38 per hour

After proposed crew, μ = 10 Total hourly cost = 6 x 4 + [1/(10 – 4)] x 2 x 13 = ` 28 per hour (approximately) Therefore, the total hourly cost after the addition of one more crew of three persons is less than the existing cost. Thus, the truck owner should add a crew of 3 loaders to reduce the total hourly cost. 11.8 Multiple Server Model: (M/M/S): ( /FCFS) In the case of multiple servers with multiple queues, the system has several servers with various queues, all of which are processing the request of the customers. The multiple server model is applicable for queuing models with two or more serving channels. The most common multiple-channel system contains parallel service channels serving a single queue. In this model, it is assumed that the customers arrive according to the Poisson process, at an average rate of λ customers per unit of time , and are served on FCFS basis. These servers are identical and each customer is served according to the

exponential distribution with an average rate of μ customers per unit of time. It is assumed that only one queue is formed. The overall service rate of servers is obtained in two situations, when there is ‘n’ number of customers in the system, where: n = Number of customers k = Number of service stations or servers

If n< s, then there will be no queue. The combined service rate will be μ n = nμ; n
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



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The arrival is infinite A single waiting line is formed



The service is given on FCFS basis



The arrival of customers follows the Poisson probability law and service time has an exponential distribution

The following are the operating characteristics of the multiple server model:

λ = Average rate of arrivals μ = Average service rate of each of the servers kμ = Mean combined service rate of all the servers ρ = λ/ kμ = Utili sation factor of the entire system 1. Probability that the system shall be idle is given by:

P 0 2.



( )

P (n



n

n!

≥ k)

.P o



, when n ≤ k, and

k (  ) k



k !k 

. P o

Pn



(  )n



k !k n k

.P o

when n > k, and

(probability that a patient has to wait)

The expected number of customers waiting in the queue:

Lq 4.



(  ) k  k !(1   )

Probability that there shall exactly be n customers in the system is Pn

3.



k 1 n0

(  ) n  n!



( /  ) k  P 0 k (1   ) 2

k

or

 ( /  ) P 0 (k  1)!(k    ) 2

The expected number of customers in the system: Ls = Lq + (λ/ μ)

5.

The average time a customer spends in the queue waiting for service: Wq = Lq / λ


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The expected waiting time that a customer spends in the system: Ws = Wq + 1/ μ

Some of the depictions of multiple servers with multiple queues are given in Figure 11.6:

Figure 11.6: Multiple Queues with Multiple Servers

Figure 11.7 depicts another example, wherein initially the queue was single but later it got converted into multiple queues:

Figure 11.7: Another Fo rmat of Multiple Queue with Multiple Server

Thus, from the managerial perspective, the manager is required to manage this system in an efficient manner. Example 4: ABC service station has 5 mechanics who can service a scooter in 2 hours on an average. The scooters arrive at the service station at an average rate of 2 scooters per hour. Assuming that scooter arrivals


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are Poisson distributed and the servicing times are distributed exponentially, determine, a.

Utilisation factor

b.

The probability that the system shall be idle

Answer : λ = 2 scooters/hour

μ = ½ scooter/hour k=5 a.

Utilisation factor = ρ =

λ/kμ

= 2/(5*1/2) = 0.8 b.

The probability that the system shall be idle:

 n  (  ) k   k 1 (  )  P 0    n 0   n! k !(1   )    

1

n k  (2 ) ( 2 )  51   . 5 . 5 = P 0   n 0   n! 5!(1  .08)   

1

= 103/3 +128/3 = 77-1 11.9 Self Service Model: (M/M/∞): (GD/∞/∞) There are some situations wherein the customer is never required to wait for the service to begin. No matter what time the customer arrives, there is always someone to process her/his request. These types of systems are called infinite server or self-service model. An example is the widely used Interactive Voice Response System (IVRS) that is available in most of the customer care services. In this case, the customer is never required to wait for availing the services. No matter what time he/she arrives, her/his request is processed. Figure 11.8 depicts a picture of the self-service model:


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Figure 11.8: Self-Service Model

Self-service models are easier to design and apply, but they are limited to the constraints of taking their own de cision. In the generalised self-service model, we have:

λn = λ, n = 0, 1, 2… μn = nμ, n =0, 1, 2, … Thus, Pn = (λn / n! μn) P0 = ρn/n! P0, n = 0, 1, 2… Because, P0



 n 0 P n  1 , it follows that 1

1  



2

  .......

/ 2!



1 e

 e 

As a result, Pn



e n n!

n  0,1,2..........

Here, the mean Ls = ρ and L q and W q are 0. Example 5: An investor invests ` 1,000 every month in a stock market. He usually keeps the securities for about 3 years and sells them when an opportunity appears. It was observed that about 25% of the securities


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decline at about 20% a year and the remaining 75% appreciated at a rate of 12% a year. Determine the investor’s average equity in the stock market. Solution: This is a self-service model because the investor does not have to wait in queue to buy or sell securities. Average time between order placements is 1 month

λ = 12 securities per year μ = Rate of selling security = 1/3 security per year ρ = λ/μ = 36 securities The estimate of the long run average annual net worth of the investor is: (.25 * 1000)(1-.20) + (.75*1000)(1+.12) = ` 63,990

11.10 Machine Serving Model: (M/M/R) R
nμΔt when n≤ R



RμΔt when n≥ R

On the other hand, the probability of a single arrival during Δt is: (K-n)λΔt when n≤K

Where λ = Rate of breakdown per machine Thus,

 K  n P n     P 0 , 0≤n≤R or n


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 K  n!  n P P n    , R  n  K n  R n R ! R  0 Where

P n

 K  n P n     P 0 , 0≤n≤R or n

 K  n!  n P , R  n  K   n  R n R ! R  0

The average number of broken machines is represented as:

Lq



K

 (n  R) P n

n  R 1

The average time a broken machine needs to wait to get serviced is calculated as follows:

W q = Lq/ λeff ,

system, arrivals occur with the rate λ, but all arrivals do not join the system. Therefore, λeff includes only those arrivals that join the In the queuing system.

Suppose λeff defines the effective arrival rate, then the value of λ eff can

be

conveniently determined from the following equation:

 eff   

k

 ( K  n) Pn n 0

Let us study the machine serving model with the help of an example. Example 6: XYZ organisation manufactures products using five machines. Each of the machines (when running) suffers break downs at an average rate of 2 per hour. There are 2 servicemen and only one man can work on a machine, at a time. If n machines are out of order when n>2, then (n-2) of them wait until a serviceman is free. After a service man starts work on a machine, the time to complete the repair has an exponential distribution with a mean of 720 minutes. Find the distribution of the number of machines out


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of action at a given time. In addition, find the average time an out-of-action machine has to spend waiting for the repair to start. Solution: Given that

Average rate of break downs λ = 2 and μ = 720/60 = 12 K = Total number of machines in the system = 5 R = Number of servicemen = 2

ρ = λ/μ = 2/12 Let n = Number of machines out of order

 5 n (2 / 12) P 0 ,0  n  2   n or

Pn  

5 n!(2 / 12)n P 0 ,2  n  5   n 2 n 2 ! 2   2

5

∑

P0= n=0

n!(2/12) n

5

(2/12) n + ∑ n

-1

= 648/1493 n=3

2! 2n-2

Substituting this value of P0 in Pn, we get

5  n!(2 / 12) n P 0 P n    ,2  n  5 n 2 n 2 ! 2  0 1

 2  5 n!( 2 / 12) n  n 5 P 0   n0   ( 2 / 12)   n3   648 / 1493 n 2 n 2 ! 2     Average number of machines out of action is given by: Lq =

 5n21


(n-2) Pn

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Unit 11

 5n 3 (n-2) Pn = P3 + 2P4 + 3P5 = 165/1493

Average time an out-of-action machine has to spend waiting for the repairs to start is W q = Lq/ λeff

λeff = λ *  5n 3 (5-n) Pn = λ (5P0+4P1+3P2+2P3+P4) = 12300/1493 (since λ=2) Wq = (165/1493)* (1493/12300) = 55/4100 hrs = 33/41 minutes Activity 1:

1. In a single server system, the arrival rate, λ = 5 per hour and service rate, μ = 8 per hour. Find out the probability that the server is idle and the probability that there are at least two customers in the system. 2. If the arrival rate is 25 per hour and service rate is 30 per hour. Calculate the mean waiting time. Objective of the activity: To help you explore queuing problems Feedback: Hint: Answer 1. 15/64; (5/8)3 Answer 2. 10 minutes Self Assessment Questions: 12. Why is the single server model used? 13. In case of single server several queues, there is no queue. (true/ false) 14. The ___________ is applicable for queuing models having two or more serving channels. 15. In ______________ the customer is never required to wait for availing the services. No matter what time he/she arrives, his/her request is processed. 16. Which of the following model involves one queue and one service counter?


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a. Single server single queue b. Single server several/multiple queues c. Multiple server model d. Duplicate server model 17. Which of the following models has an assumption of a single waiting line with infinite capacity? a. Multiple server model b. Self-service model c. Single server model d. Machine serving model

11.11 Summary Let us recapitulate the main points discussed in the unit: 

In a queue model, there is a server to process the request placed by the customers standing in the queue.



Pure birth model is a model in which the queue keeps on getting added with new customers.



Pure death model is a model in which there is no addition to the queue but only those who are already in the queue are required to be served.



A queue may have single server with single queue or single server with multiple queues.



Queues follow a Poisson distribution with exponential growth.

11.12 Glossary Let us have an overview of the important terms mentioned in this unit: Arrival pattern: It is the average rate at which the customers arrive.


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Balking: It is the phenomenon that is commonly witnessed when customers do not want to join the queue on account of the length of the queue or because they have changed their decision. Collusion: It is the phenomenon that is observed when one person joins the queue and avails the service on behalf of others. Jockeying: It is the phenomenon when customers keep jumping the queue every now and then. Pure death model: It is a model in the queuing theory in which there is no addition to the queue length but only those who are standing in the queue will be served or are being served. Queue discipline: It is the way in which service providers attend customers waiting in the queue. Reneging: This is the phenomenon that is observed when people in a queue wait till a certain period of time and then quit on remaining unattended. Service facility: It is the facility that is provided to customers when they stand in the queue. Single server: It is a type of queue wherein there is a single person serving the persons standing in the queue.

11.13 Terminal Questions 1.

What do you mean by queuing?

2.

Explain the applications of the queuing model.

3.

Discuss the pure birth model in detail.

4.

Write a short note on specialised Poisson queues.

5.

Explain the multiple server model of the queuing system.


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11.14 Answers Self Assessment Questions 1.

Queues are waiting lines that are generally seen at banks, railway ticket counters, temples, clinics, petrol pumps, and bus stop s.

2.

True

3.

a. Output source

4.

c. Research

5.

The pure birth model is used for estimating the arrival time distribution of customers. This model is based on the assumption that the customers who arrive at the queuing system do not leave the queue without receiving service.

6.

Markovian Property of inter-arrival time

7.

a. For estimating the arrival time distribution of customers

8.

Pure death model

9.

True

10. True 11. First come, first served 12. The single server model helps in solving the queuing problems with a single channel to serve customers. 13. False 14. Multiple server model 15. Self-service model 16. a. Single server single queue

17. c. Single server model


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Terminal Questions 1.

(Hint: Explain the term queuing with examples) A problem that involves waiting for a service is called a queuing problem or a waiting line problem. Refer to section 11.2 Meaning of Queuing Problems, which explains the meaning of queuing.

2.

(Hint: Define queuing and also give the different applications of the queuing models) Queuing

models

can

be

applied

to

production

processes,

transportation, communication, service industry and human resource management. Refer to sub section 11.2.1 Applications of queuing models, which explains the applications of queuing models. 3.

(Hint: Explain the pure birth model and give the conditions of the pure birth model) In the pure birth model, customers are determined to get service at any cost. Refer to sub section 11.3 Pure Birth Model, which explains the pure birth model.

4.

(Hint: Describe specialised Poisson queues and also give the formulae of the specialised Poisson queue) In specialised Poisson queues, a waiting customer is selected from the queue to start service with the first available server. Refer to sub section 11.6 Specialised Poisson Queues, which explains the concept of specialised Poisson queues.

5.

(Hint: Define the multiple server model and explain the model with assumptions and formula) The multiple server model is applicable for queuing models with two or more serving channels. Refer to sub section 11.8 Multiple Server Model, which explains the concept of the multiple server model.


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11.15 Case Study: Queuing Problem in a Restaurant In the take-away section of a restaurant, customers arrive at one window and drive according to Poisson distribution with a mean of 10 cars per hour. The service time per customer is exponential with a mean of 5 minutes. A maximum of 3 cars can be accommodated in the space in front of the window. All other cars wait outside in an indicated space. Discussion questions: 1.

Find out the probability of an arriving customer driving directly to the space in front of the window. (Hint: 0.42)

2.

Find out the probability of an arriving customer having to wait outside in the indicated space. . (Hint: 0.48)

3.

How long is an arriving customer expected to wait before service starts. (Hint: 0.417)

References and Suggested Readings 

D. Gross, & Harris, C. (2008). Fundamentals of Queuing Theory (3rd ed.)



Chitale, R. (2008). Probability and Queuing Theory . Technical Publications.



Haviv, M. (2013). A course in Queuing Theory . Springer London Limited.

E-References 

People.brunel.ac.uk (1908). Queueing theory. [online] Retrieved from: http://people.brunel.ac.uk/~mastjjb/jeb/or/queue.html [Accessed: 1 Jul 2013].



Eventhelix.com (n.d.). Untitled. [online] Retrieved from: http://www.eventhelix.com/realtimemantra/.../queueing_theory.htm [Accessed: 1 Jul 2013].



Unknown. (n.d.). Untitled. [online] Retrieved from: http://www.win.tue.nl/~iadan/queueing.pdf [Accessed: 3 Jul 2013].


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