Erlang distributed activity times in stochastic activity networks

SANs are assumed a priori cumulative distribution function (c.d.f.). Normally distributed is proposed by Sculli [15] and Kamburowski [7], exponentially distributed is assumed by Kamburowski [8], Kulkarni and Adlakha [10] and Magott and Skudlarski [12]. Bendell et al [1] have considered the problem of using the Erlang distribution as a representation of activity times. Their method based on the moments approach which is more practical alternative to both the analytical and the numerical integration. However, they derived the rst four central moments of max(X1; X2) and X1 +X2 only where, X1 and X2 are independent r.v.'s. A survey of recent developments and complexity in SANs can be found in Elmaghraby [4] and [5]. This paper generalized the work of Bendell et al [1]. It presents the kth moments of the max(X1; X2; ; Xn) and the c.d.f. of the sum of n independent r.v.'s is also given. SANs are de ned as (N;A; F ( )), where N is the set of nodes N = f1; 2; : : : ; ng, A is a set of arcs A = fa1; a2; ; amg and F (t) = P(Tr t), for t > 0 is the c.d.f., where Tr is a r.v. which describes the duration of the arc ar. The network has one starting and one ending node and is acyclic, i. e., the nodes are numbered in such a way that whenever there exists an arc(i; j), then i < j. We shall use the names \activity" and \arc", \event" and \node", \project" and \network" synonymously. The main problem in SANs (largest and shortest path) is divided into two categories. The rst category is to nd the distribution of