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random variables each having an exponential distribution with the same mean 1/ ; see also Appendix E The Erlang probability distribution function can be numerically evaluated without using a general code for the incomplete gamma integral For xed t, the Poisson probabilities pj (t) = e t ( t)j /j ! can be recursively calculated from p0 (t) = e t and pj (t) = ( t/j )pj 1 (t) for j = 1, 2, However, exponent under ow may occur in the calculation of p0 (t) when t is very large There is a simple trick to avoid the exponent under ow De ne qj (t) = ln pj (t) The recursion scheme q0 (t) = t and qj (t) = ln( t/j ) + qj 1 (t) for j 1 offers no numerical dif culties at all Any desired pj (t) is calculated from pj (t) = exp[qj (t)] if qj (t) 100 (say) and pj (t) = 0 otherwise The trick of working with logarithms is one of the most useful tricks to avoid under ow in numerical analysis Logarithms enable us to reduce the manipulation with extremely large (small) numbers to the manipulation with moderately sized numbers The lognormal distribution A positive random variable X is said to be lognormally distributed when it has the probability density 1 1 f (t) = exp [ln(t) ]2 / 2 , t > 0, 2 t 2 where the shape parameter is positive and the scale parameter may assume each real value The probability density function F (t) equals F (t) = ln(t) , t > 0,

x where (x) = (1/ 2 ) exp( u2 /2) du is the standard normal probability distribution function The mean and the squared coef cient of variation of the lognormal distribution are given by 1 E(X) = exp + 2 2

2 and cX = exp( 2 ) 1

.

Thus a unique lognormal distribution can be tted to each positive random variable with given rst two moments The lognormal density is always unimodal with a maximum at t = exp( 2 ) The failure rate function rst increases and next decreases to zero as t and thus the failure rate is only decreasing in the long-life range The Weibull distribution A positive random variable X is said to be Weibull distributed when it has the probability density f (t) = ( t) 1 exp[ ( t) ], t > 0,

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with the shape parameter > 0 and scale parameter > 0 The corresponding probability distribution function F (t) is given by F (t) = 1 exp[ ( t) ], t 0

.

.

The mean and the squared coef cient of variation of the Weibull random variable X are 1 (1 + 2/ ) 1 2 and cX = 1+ 1 E(X) = [ (1 + 1/ )]2 A unique Weibull distribution can be tted to each positive random variable with given rst two moments For that purpose a non-linear equation in must be numerically solved The Weibull density is always unimodal with a maximum at 2 2 t = 1 (1 1/ )1/ if cX < 1 ( > 1), and at t = 0 if cX 1 ( 1) The 2 failure rate function is increasing from 0 to in nity if cX < 1 and is decreasing 2 > 1 from in nity to zero if cX 2 The gamma and Weibull densities are similar in shape, and for cX < 1 the lognormal density takes on shapes similar to the gamma and Weibull densities For 2 the case cX 1 the gamma and Weibull densities have their maximum value at t = 0; most outcomes tend to be small and very large outcomes occur only occasionally The lognormal density goes to zero as t 0 faster than any power of t, and thus the lognormal distribution will typically produce fewer small outcomes than the other two distributions This explains the popular use of the lognormal distribution in actuarial studies The differences between the gamma, Weibull and lognormal densities become most signi cant in their tail behaviour The densities for large t go down like exp[ t], exp[ ( t) ] and exp[ 1 [ln(t) ]2 / 2 ] 2 Thus, for given values of the mean and the coef cient of variation, the lognormal density always has the longest tail The gamma density has the second longest tail only if > 1; that is, only if its coef cient of variation is less than one In Figure B1 we illustrate these facts by drawing the gamma, Weibull and log2 normal densities for cX = 025, where E(X) is taken to be 1 To conclude this appendix, we discuss several useful generalizations of exponential and Erlangian distributions In many queueing and inventory applications there is a very substantial (numerical) advantage in using the generalized distributions rather than other distributions Generalized Erlangian distributions An Erlang-k (Ek ) distributed random variable can be represented as the sum of k independent exponentially distributed random variables with the same means A generalized Erlangian distribution is one built out of a random sum of exponentially distributed components A particularly convenient distribution arises when these components have the same means In fact, such a distribution can be used to approximate arbitrarily closely any distribution having its mass on the positive.

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