# The influence of the form of the probability distribution of the replenishment lead time on the performance of an inventory node

It is not surprising that even in cases with identical average replenishment lead times, differences in the form of the probability distribution may have substantial effects on the performance of an inventory node and thus on the required inventory level. In order to demonstrate this numerically, consider an $(s,q)$ policy with continuous review and Poisson demand that arrives at a rate $\lambda$ on a continuous time axis. The system parameters are: $\lambda=2, q=5, s=9$. The demand size is always one unit.

We analyze this inventory node with different probability distributions of the lead time, all with identical expectations $E\{L\}=4$, with the help of a simulation model. In addition to the case with deterministic lead time $\ell=4$ (Case 1), we consider the lead time distributions shown in the following Table.

 Case 2 Case 3 Case 4 $\ell$ $P\{L=\ell\}$ $\ell$ $P\{L=\ell\}$ $\ell$ $P\{L=\ell\}$ 3 0.3 2 0.2548 2 0.35 4 0.4 3 0.2162 3 0.24 5 0.3 4 0.1776 4 0.13 5 0.1390 5 0.05 6 0.1005 6 0.05 7 0.0619 7 0.05 8 0.0500 8 0.05 9 0.05 10 0.02 11 0.01 $E\{L\}$ 4 4 4 $Var\{L\} 0.60 3.13 5.60 The performance is measured with the$\beta$service level. The following Table summarizes the simulation results.  Case 1 2 3 4$\beta\$ 86.5% 84.4% 81.1% 79.9%

It appears that the randomness of the replenishment lead time may have a significant influence on the performance of an inventory node. This negative influence increases with the coefficient of variation of the lead time. If it is not possible to eliminate the causes of the lead time variations, then safety stock is required.

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