1. MATS-30004 Quantitative Methods Dr Huw Owens

2. Queuing Models Imagine the following situations: ……
Shoppers waiting in front of a check-out stand in a supermarket;
Cars waiting at a red traffic light;
Patients waiting in a lounge of a medical centre;
Planes waiting for taking off in an airport;
Customers waiting for tables in Pizza Hut (especially at weekends); ……
What these situations have in common is the phenomenon of waiting.
Waiting causes inconvenience, but like it or not, it is part of our daily life. All we hope to achieve is to reduce the inconvenience to bearable levels.
29/11/2018

3. The waiting phenomenon is the direct result of randomness in the operation service facilities.
In general, the customer’s arrival and service time are not known in advance.
Our objective in studying the operation of a service facility under random conditions is to secure some characteristics that measure the performance of the system under study.
29/11/2018

4. Some of the characteristics that are of interest are the following:
The average number of customers in the queue
The average number of customers in the system (the number in queue + the number being served)
The average time a customer spends in the queue
The average time a customer spends in the system (the waiting time plus the service time)
The probability that an arriving customer has to wait for service
The probability of n customers in the system
Managers who have the above information are better able to make decisions that balance desirable service levels against the cost of providing the service.
29/11/2018

5. The Structure of a Queue
As customers arrive at a facility, they join a queue.
The server chooses a customer from the queue to begin service. Upon the completion of a service, the process of choosing a new waiting customer is repeated.
This is how a queuing situation is created. It is assumed that no time is lost between the completion of a service and the admission of a new customer into the facility. A simple queuing system is illustrated in Figure 1.
This diagram depicts a single channel queue, i.e., all customers entering the system must pass through the one channel.
When more customers arrives at the facility, It is necessary the system have multiple channel queue to serve the customers.
29/11/2018

6. Obviously, the main “actors” in a queuing system are the customers and the server.
The elements that are of most interest in the queuing system are the pattern in which the customers arrive, the service time per customer, and the queue discipline.
System
Customer leaves
Customer arrivals
Queue
Server
Figure 1. A single channel queue system
29/11/2018

7. The Process of Arrivals
Defining the arrival process for a queue involves determining the probability distribution for the number of arrivals in a given period of time.
In many queuing situations the arrivals occur in a random fashion; that is, each arrival is independent of other arrivals, and we cannot predict when an arrival will occur.
In such cases, the Poisson probability distribution is used to describe the arrival pattern.
Using Poisson probability function, the probability of x arrivals in a specific time period is defined as follows:
(for x = 0, 1, 2…) (1)
where
x = the number of arrivals in the period
λ = the average or mean number of arrivals per period
e =
29/11/2018

8. Example 1
The North-west Fried Chicken has analysed data on customer arrivals and has concluded that the mean arrival rate is 45 customers per hour.
For a 1-minute period, the mean number of arrival would be λ = 45/60 = 0.75 arrivals per minute.
We can use the following probability function to compute the probability of x arrivals during a 1-minute period:
The Poisson probability distribution during a 1-minute period is listed as follows:
29/11/2018

9. The Distribution of Service Times
The service time is the time the customer spends at the service facility once the service has started.
Service time normally varies according to the individual situations.
In the North-west Fried Chicken example, each customer’s order may be of different sizes. It may take 1 minute to fill a small order, but 2 or 3 minutes to fill a larger order.
It has been determined that the exponential probability distribution often provides a good approximation of service times in queuing situations.
If the probability distribution for service times follows an exponential probability distribution, the probability that the service time will be less than or equal to a time of length t is given by
P(service time  t) = (2)
where
 = the average or mean number of customers that can be served per time period
29/11/2018

10. Example 1
Suppose that North-west Fried Chicken has studied the order-taking and order- filling process and has found that the single server can process an average of 60 customer orders per hour.
On a 1-minute basis, the average or mean service rate would be  = 60/60 = 1 customer per minute. This makes the equation (2) become
P(service time  t) = for this case.
For example
P(service time  0.5 min.) = 1 - e-1(0.5) = =
P(service time  1.0 min.) = 1 - e-1(1.0) = =
P(service time  2.0 min.) = 1 - e-1(2.0) = =
29/11/2018

11. Queue Discipline
In describing a queuing system, we must define the manner in which the queuing customers are arranged for service. The possible queue disciplines are:
First-in-first-out (FIFO) or first-come-first-served (FCFS)
Last-in-first-served (LIFO)
Service-in-random-order (SIRO)
Others
The FIFO or FCFS queue discipline applies to many situations.
29/11/2018

12. Steady-State Operation
A queuing system gradually builds up to a normal or steady state from its beginning.
The start-up period is known as the transient period.
This period ends when the system reaches the normal or steady-state operation.
Queuing models describe the steady-state operating characteristics of the queue.
29/11/2018

13. The Single-channel Queue Model with Poisson Arrivals and Exponential Service Times
This is the simplest queuing model and it assumes the following:
The queue has a single channel
The pattern of arrivals follows a Poisson probability distribution
The service times follow an exponential probability distribution
The queue discipline is FCFS
The formulas that can be used to develop the steady-state operating characteristics in this situation are given below, with
 = the mean or average number of arrivals per time period (the mean arrival rate)
 = the mean or average number of services per time period (The mean service rate)
29/11/2018

14. P0 = 1 - / (3) Lq = 2/[(-)] (4) L = Lq + / (5) Wq = Lq/ (6)
The probability that there are no customers in the system
P0 = 1 - / (3)
The average number of customers in the queue
Lq = 2/[(-)] (4)
The average number of customers in the system (the number in queue + the number being served)
L = Lq + / (5)
The average time a customer spends in the queue
Wq = Lq/ (6)
The average time a customer spends in the system (the waiting time plus the service time)
29/11/2018

15. W = Wq + 1/ (7) Pw = / (8) Pn = (/)n P0 (9)
The probability that an arriving customer has to wait for service
Pw = / (8)
The probability of n customers in the system
Pn = (/)n P (9)
Formula (8) provides the probability the service facility is busy, and thus the ratio / is often referred to as the utilisation factor for the service facility.
The above formulas are applicable only when the mean service rate  is greater than the mean arrival rate , i.e., when / < 1.
If this condition does not exist, the queue will grow without limit since the service facility does not have sufficient ability to handle the arriving customers. Therefore, to use these formulas, we must have  > .
29/11/2018

16. Example 1
Recall that for the North-west Fried Chicken problem we had a mean arrival rate of  = 0.75 customers per minute and a mean service rate of  = 1 customers per minute. Obviously the condition / < 1 is satisfied. Then we can use the formulae to calculate the operation characteristics for North-west Fried Chicken single queue:
P0 = 1 - / = /1 = 0.25
Lq = 2/[(-)] = 0.752/[(1(1-0.75)] = 2.25 customers
L = Lq + / = /1 = 3 customers
Wq = Lq/ = 2.25/0.75 = 3 minutes
W = Wq + 1/ = 3 + 1/1 = 4 minutes
Pw = / = 0.75/1 = 0.75
29/11/2018

17. Equation Pn = (/)n P0 can be used to determine the probability of any number of customers in the system. Some computation results are listed in the following table.
29/11/2018

18. Looking at the results of the single-channel queue for the North-west Fried Chicken, we can learn several important things about the operation of the queuing system:
Customers wait an average of 3 minutes before beginning to place an order, which is somewhat long for a business based on fast service;
the average number of customers waiting in line is 2.25 and 75% of the arriving customers have to wait for service; this indicates something needs to be done to improve the efficiency of the queue operation;
There is a probability that seven or more customers are in the queuing system at one time, which indicates a fairly high probability that the North-west Fried Chicken will periodically experience some very long queues if it continues to use the single channel operation.
29/11/2018

19. If the operating characteristics are unsatisfactory in terms of meeting companies standards for service, something will need to be done to improve the queue operation.
To reduce the waiting time mainly means to improve the services rate. The service improvements can be made by either
increasing the mean service rate  by making a creative change or by employing new technology; or
adding parallel service channels so that more customers can be served at a time.
29/11/2018

20. Example
Suppose that the North-west Fried Chicken management decides to increase the mean service rate  by employing an order filler who will assist the order taker at the cash machine.
By this change, the management estimates that the mean service rate can be increased from the current 60 customers/hour to 75 customers/hour.
The mean service rate for the revised system is  = 75/60 = 1.25 customers per minute.
Then using  = 0.75 and  = 1.25, the characteristics of the revised system are calculated as follows:
29/11/2018

21. Comparison between the original system and the revised system shows the service of the North-west Fried Chicken has been greatly improved due to the action taken.
29/11/2018

22. The Multiple-channel Queuing Model with Poisson Arrivals and Exponential Service times
A multiple-channel queue consists of two or more channels or service locations that are assumed to be identical in terms of service capability.
In the multiple channel system, arriving customers wait in a single queue and then move to the first available channel to be served (such as the queue waiting to pass the security check in Manchester airport).
A 3-channel queuing system is depicted in Figure 2.
29/11/2018

23. 29/11/2018 Figure 2. A three channel queue system System
Server1
Customer leaves
System
Queue
Figure 2. A three channel queue system
Server3
Server2
Customer arrivals
29/11/2018

24. Assumptions for the queue in the multiple-channel system:
The queue has two or more channels
The pattern of arrivals follows a Poisson probability distribution
The service time for each channel follows an exponential probability distribution
The mean service time  is the same for each channel
The arrivals wait in the single queue and then move to the first open channel for service
The queue discipline is first-come, first-served (FCFS)
29/11/2018

25.  = the mean arrival rate for the system
Notations:
 = the mean arrival rate for the system
 = the mean service rate for each channel
k = the number of the channels
Formulas:
The probability that there are no customers in the system
(10)
29/11/2018

26. The average number of customers in the queue (11)
The average number of customers in the system
L = Lq + / (12)
The average time a customer spends in the queue
Wq = Lq/ (13)
The average time a customer spends in the system
W = Wq + 1/ (14)
29/11/2018

27. The probability that an arriving customer has to wait for service (15)
The probability of n customers in the system
for n  k (16)
for n > k (17)
29/11/2018

28. The following table is provided to simplify the use of the formulas.
While some of the formulas for the operating characteristics of multiple-channel queues are more complex than their single-channel counterparts, the expressions provide the same information.
The following table is provided to simplify the use of the formulas.
Values of P0 for multiple-channel queue with Poisson arrivals and exponential service time
29/11/2018

29. 29/11/2018

30. 29/11/2018

31. Example
Consider the North-west Fried Chicken situation again. If the management decided to add a parallel service channel to improve the service rather than to increase the mean service rate , let’s evaluate the operating characteristics for the two-channel system.
We know that k = 2,  = 0.75, and  = 1. So,
P0 = (from table)
Lq = customer
L = customer
Wq = 0.16 min.
W = 1.16 min.
Pw =
Pn7 =
29/11/2018

32. Objective function coefficients
It is clear that the two-channel system will greatly improve the operating characteristics of the queue.
29/11/2018

33. Some General Relationships for Queuing Models
Previously, we have come across a number of operating characteristics including
Lq = the average number of customers in the queue
L = The average number of customers in the system
Wq = the average time a customer spends the queue
W = the average time a customer spends in the system
John D. C. Little showed that general relationships existing among these four characteristics apply to a variety of different queues. Two of the relationships, referred to as Little’s flow equation, are as follows:
L = W (18)
Lq = Wq (19)
29/11/2018

34. Another general relationship is the expression for the average time in the system:
W = Wq + 1/
The importance of these relations is that they apply to any queuing models regardless the arrival patterns and the service time probability distribution.
29/11/2018

35. Economic Analysis of Queues
For a queuing system, the manager has to decide at what level the system should be run so that it is most economical. In other word, the manager has to make sure that the total cost for operating the queuing system is minimised.
In queuing systems, the total cost comes from two directions, i.e. the waiting cost and the service cost. Let’s use the following notations:
cw = the waiting cost per time period for each customer
L = the average number of customers in the system per time period
cs = the service cost per time period for each channel
k = the number of channels
TC = the total cost
Then the total cost is
TC = cwL + csk (20)
29/11/2018

36. A critical issue in conducting an economic analysis of a queue is to obtain reasonable estimates of the waiting cost and service cost per period.
Of these two, the waiting cost per period is usually more difficult to evaluate.
This is not a direct cost to a company. However, if this cost is ignored and the queue is allowed to grow, customers will ultimately take their business to elsewhere.
In this way, the company will experience lost sales, and incur a cost. The service cost is generally easier to determine. This cost is the relevant cost associated with operating each service channel.
29/11/2018

37. Example
Assume that in the case of Northwest Fried Chicken, the waiting cost rate is £10 per hour for customer waiting time and the service cost rate is £7 per hour. Calculate the total cost for single-channel and two-channel queuing system:
Single-channel system:
We knew that the average number of customers in the system L = 3. So
TC = £10 (3) + £7 (1) = £37.00 per hour
Two-channel system:
L =
TC = £10 (0.8727) + £7 (2) = £22.73 per hour
Thus, given the cost data provided by the North-west Fried Chicken, the two-channel system provides the more economical operation.
29/11/2018

38. The general shapes of the cost curves in the economic analysis of queues are show in Figure 3.
Total Cost per Hour
Total cost
Service cost
Waiting cost
29/11/2018

39. Other queuing Models
D G Kendall suggested a notation that is helpful in classifying the wide variety of different queuing models that have been developed. The three-symbol notation Kendall notation is as follows:
A/B/s
where
A denotes the probability distribution for the number of arrivals
B denotes the probability distribution for the service time
s denotes the number of channels
29/11/2018

40. The values for A and B that are commonly used are as follows:
M designates a Poisson probability distribution for the number of arrivals or an exponential probability distribution for service time
D designates that the number of arrivals or service time is deterministic or constant
G designates a general distribution with a known mean and variance for the number of arrivals or service time
29/11/2018

41. Using the Kendall notation, the single-channel queuing model with Poisson arrivals and exponential service times is classified as an M/M/1 model.
The two-channel queuing model with Poisson arrivals and exponential service times would be classified as an M/M/2 model.
We will not cover the other queuing models because of time limitation and the similarity of the equations for the operating characteristics. These equations can be found in the listed reference books.
29/11/2018