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)
Where
a = arrival (or interarrival) distribution,
b = departure (or service time) distribution,
c = number of parallel service channels in the system,
d = service discipline,
e = maximum number of customers allowed in the system,
f = calling source or population.
The various types of queuing models can be classified as follows:
1. Probabilistic
2. Deterministic
3. Mixed
1. Probabilistic Model
Where arrival and service rates are unknown and assumed to be random variable.
i. Poisson Distribution
A Poisson distribution is a discrete probability distribution which predicts the number of arrivals in a given time. The poisson distribution involves the probability of occurrence of an arrival. Poisson assumption is quite restrictive in some cases. It assumes that arrivals are random and independent of all other operating conditions. Poisson distribution assumes fixed time interval of continuous servicing which is never sure in all service.
2. Exponential Distribution
The most common type of distribution used for service times is exponential distribution. It involves the probability of completion of a service.
The queue theory provides a mathematical frame work to present these dimensions of a queue problem in precise statistical terms and develop a solution which can , avoiding both the extremes, meet the requirements of customers as well as service unit. The queue theory is then to minimize the total cost of queue i.e. the cost of providing service and cost of waiting time both, with the help of suitable mathematical model. Various constraints are taken into consideration in developing a queue model. There is no maximization or minimization of the objective function. Various alternatives are considered and evaluated through the queue model and a final choice of appropriate alternative and appropriate mode is made.
Model I (Erlang Model)
(M/M/1) : (∞/FCFS)
Here First M stands for Poisson arrival (exponential inter arrival time)
Second M stands for Poisson departure (exponential service time)
1 stands for single server
∞ stands for infinite capacity of the system
FCFS notation representing first come first served
Since the Poisson and exponential distributions are related to each other, both of them are denoted by the symbol ‘M’ due to Markovian property of exponential distribution.
Model II (General Erlang Model)
(M/M/1) : (∞/FCFS)
This model is same as Model I except that here the rate of arrival and service depend on the length of the line i.e. here some person interested in joining the queue may not join due to long queue and the servicing rate is also effected by the length of the queue.
Model III
(M/M/1) : (SIRO/∞/∞)
It is essentially same as Model I except that the service discipline is SIRO instead of FCFS. SIRO means service in random order.
Model IV
(M/M/1) : (FCFS/N/∞)
In this model the capacity of the system is limited or finite, say N. so the number of arrivals cannot exceed N.
Model V
(M/M/1) : (FCFS/n/M)
It is finite-population or limited source model. In this model the probability of an arrival depends upon the number of potential customers available to enter the system.
Model VI
(M/M/c) : (FCFS/∞/∞)
This is same as Model I except that there are c service channels working in parallel.
Model VII
(M/E_k/1) : (FCFS/∞/∞)
In this model except of exponential service time, there is Erlang service time with k phases.
Model VIII
(M/M/1) = GD/m/n)
where m ≤ n
it represents machine repair problem with a single repairman. N is the total number of machines out of which m are broken down and forming a queue. GD represents a general service discipline.
Model IX
(M/M/c) : (GD/m/n)
where m ≤ n
It is same as model VIII except that there are c repairmen, c<n.
Model X
This is called power supply model.
2. Deterministic Model
Model XI
(D/D)1) : (FCFS/∞/∞)
In this model interarrival time as well as service time are fixed and known with certainty. The model is , therefore, called deterministic model.
3. Mixed Queuing Model
Model XII
(M/D/1) : (FCFS/∞/∞)
Here, arrival rate is Poisson distributed while the service rate is deterministic or constant.