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≤x<∞

(5.49)Continuous exponential distribution cdfFX =1−e−λxHx

where H(x) is the step function, H(x) = 1 for x ≥ 0 and H(x) = 0 for x < 0

The means and the variance of a continuous exponential distribution are described in Eq. (5.50)

(5.50)Continuous exponential distribution meansEX=1λ andContinuous Exponential distribution varianceVX=1λ2

The continuous exponential distribution with λ = 2.5 is indicated in Fig. 5.5C.

Note: The MATLAB® code to generate the chart for Fig. 5.5 can be found in the file name: “Discrete_distributions.m,” in the directory: “…\BOOK\MATLAB_CH5_DISTRIBUTIONS.”

3.3.3.1 Exponential Distribution

Exponential distribution is widely used in electrical products, and its probability density function can be described as

(3.18)f(t)=λe−λt

where t ≥ 0. The reliability of the exponential distribution is

(3.19) R(t)=∫t∞λe−λtdt=e−λt

The reliability under exponential distribution decreases very fast with increases in the operational time, Figure 3.6(a). Hence, it is necessary to find a way to improve the reliability of electrical products.

The failure rate under exponential distribution is constant, as follows:

(3.20)λ(t)=f(t)R(t )=λe−λte−λt=λ=constant

The mean time under exponential distribution is the reciprocal of the failure rate, as follows:

(3.21)θ(MTTForMTBF)=∫0∞tf(t)dt=1λ

There is a very important characteristic in exponential distribution—namely, memorylessness. It can be described with the following conditional probability:

(3.22)P{(T>t0+t )|(T>t0)}=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

This is seen to be independent of t. The conditional probability that the job is completed within a further s seconds is given by the very same exponential distribution as would be used to determine the probability of the job being completed in s seconds starting from time 0. Often it is said that the job is being processed at rate φ, meaning that the probability that the job is completed in the next infinitesimal time interval dt is φ dt, given it has not been completed already.

The Poisson process may be constructed from the exponential. Indeed, it is useful to regard the Poisson process as a sequence of customers arriving at a queue with mutually independent and identical exponential interarrival times. As such, the Poisson process ‘inherits’ the memoryless property from the exponential. This means that the probability an arrival takes place in any given interval is independent of the history of the queue prior to that interval and in particular of the state of the queue immediately before it arrives. The parameter of the Poisson process is the same rate parameter as the underlying exponential distribution.

In fact given a sequence of independent random variable, X1, X2, X3, …. with common exponential distribution rate parameter λ, the sequence of arrival times is determined by expression 3.3, starting from timed 0.

(3.3)X1,X1+X2,X1+X2+X3,……..

Now that the construction follows, the Poisson distribution can be checked. To do this recall that the density of the sum of k independent exponential random variable with rate parameter λ is given by expression 3.4, which is the k-Erlangian distribution with parameter λ (Feller, 1970).

(3.4)λ(λt)k−l(k−l)!e−λt

Denote the number of arrivals in time interval (0,t] by Nt, then Equation 3.5 may be obtained.

(3.5)Pr{Nt=k}=Pr{X1+…+Xk≤t; X1+……+Xk+Xk+l>t}=∫0tλ(λs)k−l (k−l)!e−λse−λ(t− s)ds=(λt)kk!e−λt

4.4.4 The Generalized Exponential Distribution

The generalized exponential distribution is a three-parameter function, given by equations (46) and (47), where m, k and y are constants. In the case where m = 1 and y = 0.5 this is the conventional χ2 distribution. It becomes the most probable distribution for k = m = 1, the Schulz exponential distribution for m = 1 and the log-normal distribution for m = 0. With the availability of computers, fitting of the three-parameter equation to experimental data has become more feasible and more popular. Some of the problems in doing so have been discussed by Jakes and Saudek.18

(46)

(47)

4.4 Poisson Distribution of the Number of Arrivals and the Exponential Distribution of Headways

Depending on the traffic flow conditions, various distributions could be used to describe headways. Practically, there is no interaction between the arrivals of any two vehicles in the case of low traffic volumes. In this case, vehicles randomly show up and pass. The time interval between the appearances of successive vehicles (headway) could be, for example, 5, 10, 11, 14 s, etc. In other words, the time interval between vehicle arrivals is a random variable that frequently has exponential distribution (Fig. 4.5).

The probability density function in the case of exponential distribution equals

(4.6) fx=λe−λx

where λ > 0 is the average arrival rate.

The average arrival rate λ is expressed in vehicles per second. Since flow q is expressed in vehicles per hour, we can write the following:

(4.7)λ=q3600

The exponential distribution can be derived from the distribution of the number of vehicles that appear during specified time interval. If the distribution of the number of vehicles that appear during specified time interval is Poisson distribution, the exponential random variable will represent the time between two successive vehicles. Poisson process is characterized by the following four postulates:

The probability that at least one vehicle arrives during a small time interval Δt is approximately equal to λ Δt. (For example, when flow of vehicles q equals 360 veh/h, the probability that at least one vehicle arrives during a small time interval Δt = 3 s equals (360 veh/3600 s) × 3 s = 0.3.)

The number of vehicle arrivals in any prespecified time interval does not depend on the starting time point of the interval. The number of vehicle arrivals also does not depend on the total number of vehicle arrivals observed prior to the interval.

The numbers of vehicle arrivals in disjoint time intervals are mutually independent random variables. (The number of vehicle arrivals between 9:00 am and 9:05 am does not depend, for example, on the number of vehicle arrivals between 10:05 am and 10:10 am.)

Two or more vehicle arrivals cannot happen simultaneously.

The probability P(k) that the total number of vehicle arrivals happening in a time interval of the length t is equal to k is calculated in the case of Poisson process as:

(4.8)Pk=λtke −λtk!

Let us consider the case when no vehicles arrive in a time interval t. The probability of this event equals

(4.9)P0=λt0e−λt0!=e−λ t

We conclude that if no vehicles arrive in a time interval t, the headway h is equal to or greater than t, ie,

(4.10)P0=Ph≥t =e−λt

(4.11)1−Ph

(4.12)Ph

(4.13)Ft=1−e−λt

(4.14)ft=ddtFt=ddt1−e−λt=λ⋅e−λt

In this way, we proved that the exponential random variable represents the time between two successive vehicles in the case when the distribution of the number of vehicles that appear during specified time interval is Poisson distribution.

7.7.4 Exponential Distribution

For the exponential distribution, the characteristics hazard rate z, failure density f, reliability R, and failure distribution F have been derived above, and are:

[7.7.12]z=λ

[7.7.13]f=λexp( −λt)

[7.7.14]R=exp(−λt)

[7.7.15]F=1− exp(−λt)

for the range 0≤t≤∞. These quantities are shown in Figure 7.4(a). The distribution is characterized by a single parameter, the hazard rate λ.

The exponential distribution, which has a constant hazard rate, is the distribution usually applied to data in the absence of other information and is the most widely used in reliability work.

4.3.2 Lognormal Distribution

Shortcomings in the exponential distribution function have prompted the use of alternative distribution functions to model reliability data. One of the most popular of these is the lognormal distribution function. Mathematically, the lognormal is not a separate distribution, because by taking natural logarithms of all data points, the transformed data can be analyzed as a normal distribution. The lognormal PDF is given by

(4.10)f (t)=1tσ(2π)1/2exp(−[(ln(t)−ln(μ)]22σ2),

while the lognormal CDF has the form

(4.11)F(t)=Φ[σ−1ln(tμ)].

Here Φ(z)=12[1+Erf(z/21/2)] or 12[2−Erfc(z/21/2)], since Erfc(z/21/2) = 1 − Erf(z/21/2).

Substitution in Eqn (4.8) shows that the hazard function is given by

(4.12)λ(t)= 21/2exp[−1/2σ2ln( t/μ)2]π1/2tσErfc[ln(t/μ )/σ(1/2)1/2].

In these expressions the error function (Erf) and complementary error function (Erfc) are related to the integral of the Gaussian function and appear in connection with solutions to diffusion problems (see Eqn (5.4)). Both range from 0 to 1, while their arguments assume any values. Comparison with Eqn (4.1) reveals that the lognormal distribution is derived from the normal distribution by substituting ln t for x. Just as there are two parameters that define the normal distribution, two parameters also define the lognormal distribution. The first is μ, which has the physical significance of representing the median time, or time when 50% of the distribution will fail, i.e., μ = ln t50. Second, there is the shape parameter σ, which strongly influences the shape of f(t) and F(t), as shown in Figure 4.11(a) and (b). Note that σ is not the standard deviation of a population of lifetimes as in the normal distribution. Also shown in Figure 4.11(c) is the variation in lognormal failure rates as a function of σ.

Later, in Section 4.3.5, it is shown how to extract σ values from reliability plots. Large values of σ (σ > 2) reflect initially high failure rates that decrease with time. When a semiconductor product is in early manufacture, and quality control is still a problem, typical values of σ are as high as 3 or 4. Such values are indicative of processing and failures that are out of control. Later, when manufacturing has advanced along the learning curve and the product is a candidate for high-reliability applications, σ ≈ 1 or less [12]. When σ is close to 1, the failure rate is roughly constant, but it increases for values of less than ∼0.5. Thus the lognormal distribution can represent the early failure, steady state, or wear-out time regimes in the life of devices. As an example, failure times of GaAs–AlGaAs lasers aged at 70 °C are plotted in lognormal fashion in Figure 4.12.

A useful and widely employed way to display the hazard rate for lognormal distributions, known as the Goldthwaite plot, is shown in Figure 4.13 [14]. Based on Eqn (4.12), this representation plots the failure rate (in units of FITs × t50) versus the normalized time t′ (a ratio of t/t50) for different values of σ. It is apparent that an increase in σ has the effect of moving the failure-rate peak to shorter times; that is why small σ values are desired. Superimposed on the figure are plots of the locus of constant failure rate (λo) at a given time to. Thus the ordinate λ t50 is taken as λo t50, so that λ t50 = (λo to)/(to/t50). This equation plots as a straight line with a slope of −1 in the log–log plot of Figure 4.13. The interested reader should determine what sample size and data could be used to measure t50 = 108 h and a life of 40 years. Often projections are far more precise than we can measure. Are they then accurate? What confidence is there in their use?

As an application [16] of this figure, consider a failure distribution with σ = 2. The failure rate is low initially, but it rises with time. When t/t50 ≈ 3 × 10−3, the failure rate is 10 FITs, and t is equal to 40 years. For this to happen t50 must be equal to 1.2 × 108 h; if t50 is less than this value, the failure rate will increase to more than 10 FITs in 40 years.