Queuing System
Business of all types, industries, schools, hospitals, cafeterias, book stores, libraries, banks, post office, petrol pumps, theatres – all have queuing problems.
Queues are also found in industry – in shops where machines wait to be repaired, in tool cribs where mechanics wait to receive tools.
Waiting line problems arise either because
- There is too much demand on the facilities, so that we can say there is an excess of waiting time or inadequate number of service facilities
- There is too less demand, in which case there is too much idle facility time or too many facilities.
In either case, the problem is to either schedule arrivals or provide proper number of facilities or both so as to obtain an optimum balance between the cost associated with waiting time and idle time.
Major constituents of a queuing system
Customer: The arriving unit that requires some service to be performed. The customers may be Persons, machines, vehicles, parts etc.
Queue (Waiting line): The number of customers waiting to be serviced. The queue does not include the customers being serviced.
Service channel: The process or facility which is performing the service to the customer. This may single or multi channel. The number of service channels is denoted by the symbol c
Elements of a Queuing System
- Arrival distribution (Input or arrival (inter – arrival) distribution) - Poisson distribution
- Service distribution (Output or departure (service) distribution) - exponential distribution
- Service Channels
- Service disciplines -FCFS / LCFS / SIRO
- Maximum number of customers allowed in the system
- Population (Calling Source)
- Customer’s behavior
Operating Characteristics of a Queuing System
Length of the system, πΏπ - The average number of customers in the system including those waiting as well as those being served.
Length of the queue,πΏπ - The average number of customers in the queue waiting to get service. This excludes the customers being served.
Total time in the system / waiting time in the system ππ - The average total time spent by a customer in the system from the moment he arrives till he leaves the system. It is taken to be waiting time plus the service time.
Waiting time in the queue ππ- the average time for which the customer has to wait in the queue to get service.
Kendall’s Notation for Representing Queuing Models
D. G. Kendall (1953) and later A. Lee (1966) introduced useful notation for queuing models.
The complete notation can be expressed as
(a/b/c) : (d/e/f)
Symbols for a and b
M = Markovian (Poisson) arrival or departure distribution.
πΈπ = Erlangian or gamma inter arrival or service time distribution with parameter k
GI = general independent arrival distribution,
G = General departure distribution
D = deterministic inter arrival or service times.
Symbols for d
FCFC = First come, first served
LCFS = Last come, first served
SIRO = Service in random order
GD = General service discipline
Single Server, Infinite Queue Length Models
M- Arrival (Input or arrival (inter – arrival) distribution - Poisson distribution); Arrivals at the rate of Ξ» / hour
M – Service (service distribution - exponential distribution); Serves at the rate of ΞΌ/ hour
1 – Number of servers
∞ – Queue length (maximum number of customers allowed in the system)
∞ – Population (calling source)
M represents memory less property of arrival and service or Markovian properties
Memory Less Property
In the memory less property, we are going to assume that the behavior of the system does not have any memory.
Therefore, it does not take into account the earlier states of the system in order to define the present state of this system.
The memory less property also helps us in one important property which is an important assumption here, that during a very small interval only one event will take place.
One event takes place in a small interval.
An event is either an arrival or a service.
Example: M/M/1:FCFS/∞/∞
Consider the arrivals at the rate of 8/hr and service at the rate of 9/ hr.
Find out the following:
RO (Ο): overall system utilization
P0: the probability that all servers are idle
Lq: long-run average number of customers in the queue
Wq: long-run average time spent in the queue
X: system throughput
L: long-run average number of customers in the system
W: long-run average time spent in the system
Wqq: long-run average time spent in queue when there is queue in a queueing model
Lqq: long-run average number of customers in queue when there is queue in a queueing model
Solution:
Ξ» = 8 / hr and ΞΌ = 9/hr
Ο = Ξ» /ΞΌ = 8/9
P0 = 1-Ο = 1/9 = 0.111,
There is a probability that 11.1 % of times there is no body is in system
1- P0 = Ο = 8/9 = 0.888
Probability that at least one person in the system. Probability of server is
utilized or busy.
Probability of
there is no queue:
p (no queue) = P0 +P1 = 0.111 + ΟP0= 0.111(1+0.888) =0.2098; 21% of times there is no queue.
Using R Studio
# Import queueing package
> library(queueing)
> # Set queue model input parameters
> input_mm1 <- NewInput.MM1(lambda = 8, mu = 9, n = 0)
> # Create queue class object
> output_mm1 <- QueueingModel(input_mm1)
> # Get queue model report
> Report(output_mm1)
Output
The inputs of the M/M/1 model are:
lambda: 8, mu: 9, n: 0
The outputs of the M/M/1 model are:
The probability (p0, p1, ..., pn) of the n = 0 clients in the system are:
0.1111111
The traffic intensity is: 0.888888888888889
The server use is: 0.888888888888889
The mean number of clients in the system is: 8
The mean number of clients in the queue is: 7.11111111111111
The mean number of clients in the server is: 0.888888888888889
The mean time spend in the system is: 1
The mean time spend in the queue is: 0.888888888888889
The mean time spend in the server is: 0.111111111111111
The mean time spend in the queue when there is queue is: 1
The throughput is: 8
>
> # Get queue model summary
> summary(output_mm1)
lambda mu c k m RO P0 Lq Wq X L W Wqq Lqq
1 8 9 1 NA NA 0.8888889 0.1111111 7.111111 0.8888889 8 8 1 1 9
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