Operations Research


In queuing model, two basic constituents are considered i.e. arrival rate and service rate; these two are the main problems of waiting line.


Steady, transient and explosive state in a queue system:


1.       The distribution of customer’s arrival time and service time are the two constituents, which constitutes of study of waiting line.

2.       Under a fixed condition of customer arrivals and service facility a queue length is a function of time.

3.       As such a queue system can be considered as some sort of random experiment and the various events of the experiment can be taken to be various changes occurring in the system at any time.

We can identify three states of nature in case of arrivals in a queue system. They are :

a)      steady state,

b)      transient state

c)       explosive state.


Steady State:

The system will settle down as steady state when the rate of arrivals of customers is less than the rate of service and both are constant.

        The system not only becomes steady state but also becomes independent of the initial state of the queue. Then the probability of finding a particular length of the queue at any time will be same. Though the size of the queue fluctuates in steady state the statistical behaviour of the queue remains steady.

        steady state condition is said to prevail whenthe behaviour of the system becomes independent of time.

        A necessary condition for the steady state to be reached is that elapsed time since the start of the operation becomes sufficiently large i.e. (t → ∞), but this condition is not sufficient as the existence of steady state also depend upon the behaviour of the system i.e. if the rate of arrival is greater than the rate of service then a steady state cannot be reached. system acquires a steady state as t → ∞ i.e. the number of arrivals during a certain interval becomes independent of time. i.e.

Lim Pn (t) → Pn

t → ∞

Hence in the steady state system, the probability distribution of arrivals, waiting time, and service time does not depend on time.


Transient State

Explosive State

  •         Queuing theory analysis involves the study of a systems behaviour over time.
  •         A system is said to be in .transient state. when it’s operating characteristics or behaviour are dependent on time.
  •        This happens usually at initial stages of operation of the system, where its behaviour is still dependent on the initial conditions.
  •         So when the probability distribution of arrivals, waiting time and servicing time are dependent on time the system is said to be in transient state.
  •        In a situation, where arrival rate of the system is larger than its service rate, a steady state cannot be reached regardless of the length of the elapsed time.
  •          queue length will increase with time and theoretically it could build up to infinity. Such case is called the explosive state.