What is the importance of Poisson and exponential distribution in queuing theory?

5.1.4.2.3 Continuous exponential distribution related with successive events of discrete Poisson distribution

Continuous exponential distribution is the distance between successive events of a Poisson distribution with λ > 0. The pdf is indicated in Eq. [5.48] and the cdf in Eq. [5.49]

[5.48]Continuous exponential distribution pdf fx=λe−λxwhere0≤xt0]}=P[T>t]

Memorylessness [aka evolution without after-effects], a measure of the number of arrivals occurring at any bounded interval of time after time t, is independent of the number of arrivals occurring before time t. The proof of this relationship is follows.

Proof: According to the conditional probability, Eqn [3.22] can be expressed as

[3.23]P{[T>t0+t]|[T>t0]}=P[T>t0+t]∩P[T>t0]P[T>t0]=P[T>t0+t]P[T>t0]=e−λ[t0+t]e−λt0=e−λt =P[T>t]

Therefore, Eqn [3.22] is proved.

1.5.2.1.1 Exponential distribution

The exponential distribution has a single parameter, λ, called the failure rate, with units of inverse time.

Rt=exp−λ.tMTTF =1/λMedian=ln2/λ=ln2.MTTF=0.694MTTFBp%=1/λln11 −pscalefactor=1/λ

II.C Exponential Model

The exponential distribution is widely used in reliability. It is inherently associated with the Poisson model in the following way. If failures occur according to a Poisson model, then the time t between successive failures has an exponential distribution

[20]f[t;λ ]=λe−λtt≥0;λ>0,

where λ is the failure rate. According to Eq. [6], the failure rate function h[t; λ] = λ, which is constant over time. The exponential model is thus uniquely identified as the constant failure rate model. In other words, the failure process has no memory, which means that if the device is still functioning at time t, it is as good as new and the remaining life has the same distribution as given in Eq. [20]. Also, the exponential model is the appropriate failure model for describing the chance failure region in Fig. 1.

By a simple integration the reliability function for the exponential model is

[21]R[t;λ]=exp [−λt]

and the reliable life becomes

tR=−lnR /λ.

Also, the MTTF for the exponential model is λ−1, the reciprocal of the failure rate.

3.2.1 The memoryless property and the Poisson process

The exponential distribution is encountered frequently in queuing analysis. One reason is that the exponential can be used as a building block to construct other distributions as has been shown earlier. Indeed the distribution of virtually any positive random variable may lie approximated using the exponential [Kelly, 1979]. However, the following property is more significant reason for its importance in queuing theory.

The density of the exponential is φe−φs and integration of this gives the corresponding distribution function 1 − e−φs. Suppose X is the random variable drawn from this distribution. Say that X, “completes” when the time corresponding to X is reached starting from 0]. Suppose t seconds have elapsed and X has not completed. The memoryless property is this: that the distribution of remaining time until X completes in no way depends on t and is given by the same exponential distribution as X.

To see this must be the case, consider the following example. A random job from a class with exponential processing time requirements, with rate parameter φ, has been the sole task within a computer for t seconds. The total processing requirement is X. Let us compute the probability that the job will take at most a further s seconds to be completed. The proportion of random jobs whose processing times lie in the interval [t,t+s] is given by the area A under the graph in Figure 3.5, as marked. However this particular job is one which has taken t seconds already and it is the proportion of all such jobs which are completed within a further s seconds that is required. The probability a job takes at least t seconds is given by the tail region B. Thus the conditional distribution of the remaining time until the job is completed, given that t seconds have elapsed, is as in Equation 3.2.

[3.2]Pr{X≤t+s|t