1. WAITING LINE MODELS or QUEUEING MODELS
2. Queues are a common feature of our day-to-day life.
A queue comprises of a customer and a server.
3. Different examples of a queue
Patients waiting at the doctors clinic
Customers waiting at booking windows.
Letters to be typed at a typists desk.
Ships to be loaded or unloaded.
T.V. sets to be repaired at the repairers shop.
Phone calls arriving at the operators board.
4. Thus, queues not only comprise of people but also of goods.
5. Operation of a queueing system A customer arrives at the serving facility.
He joins the queue.
The server starts with the first customer.
Upon completing his service takes up the next customer.
This process is repeated till all the customers are served.
The time spent between the end time of the service of a customer (i.e. the departure of a served customer) and the start time of the service of a new customer is negligible.
6. Need of studying waiting time models
If the waiting time of a customer is reduced,
the idle time of the server increases.
AND IF
The idle time of the server is reduced,
the waiting time of the customers increases.
Hence the main aim is to obtain a balance between the two.
7. A queuing system can be completely described by the following features
INPUT PATTERN
SERVICE MECHNISM
QUEUE DISCIPLINE
CUSTOMERS BEHAVIOUR
MAXIMUM QUEUE LENGTH
SIZE OF THE POPULATION
8. INPUT PATTERN This explains the manner in which the customers arrive and join the system.
Generally, the customers arrive in more or less random pattern.
Hence the arrival time or the inter-arrival time is considered to be a random variable and its probability distribution is found.
9. SERVICE MECHANISM This describes the way in which the customers are served.
The service time for each customer may be different.
Hence the service time is also considered as a random variable and its probability distribution is found.
10. QUEUE DISCIPLINE This describes the way in which the customers waiting for service are chosen for service.
There are 3 different queue discipline.
FCFS (FIFO): FIRST COME FIRST SERVED
FIRST IN FIRST OUT
{e.g. queues at booking windows}
LCFS (LIFO): LAST COME FIRST SERVED
LAST IN FIRST OUT
{e.g. goods stored in go-downs}
SIRO : SERVICE IN RANDOM ORDER
{e.g. people entering the train}
11. CUSTOMERS BEHAVIOUR Generally the customers behave in the following different ways:
BALKING A customer may not join the queue because he may not like to wait or he may not have time to wait or there is no sufficient waiting space.
RENEGING A customer may leave the queue due to impatience.
JOCKEYING A customer may leave one queue and join another hoping to get service faster.
COLLUSION Several customers may collaborate and only one of them may join the queue.
PRIORITIES Some customers are given priorities over other thereby affecting the waiting times of the customers in the queue.
12. MAXIMUM QUEUE LENGTH
Sometimes only a limited number of customers are allowed in the system possibly because of space limitations. As soon as the capacity becomes full, remaining or newly arriving customers are not allowed to join the queue or are not served. Such a situation is referred to as FINITE SYSTEM CAPACITY.
E.g. Hospitals, Car-Parks, Movie Theatres, etc.
13. SIZE OF THE POPULATION
Population is also called as the calling source. Sometimes this calling source may generate finite or infinite queues.
14. To represent and describe the entire situation of queueing model the following notations called the KENDALLS NOTATIONS are used:
(M/M/1)(FCFS/8/8)
M: Inter-arrival time distribution
(Markovian/Poisson)
M: Service time distribution
(Markovian/Poisson)
1: Number of servers
FCFS: Queue discipline
8: System capacity
8: Size of the population
15. A queue denotes the number of customers waiting for their service to begin.
A system denotes the number of customers waiting for their service to begin and the one who is being served.
Hence,
The number of customers in the system =
The number of customers in queue + 1
If the number of customers in the system = n
Hence, the number of customers in the queue = n-1
16. (M/M/1)(FCFS/8/8) Notations
?: Average arrival rate
: Average departure rate
Average arrival rate ?
?: Traffic intensity = -------------------------------- = --
Average departure rate
17. Average/ Expected queue length
Lq = ?2/(1- ?)
Average/ Expected System size
Ls = ?/(1-?)
Average/ Expected waiting time in the queue Wq = ?/[(1- ?)]
Average/ Expected waiting time in the system Ws = 1/[(1- ?)]
