# First Passage Problems and Duality Relations

Chapter

## Abstract

The model of a system {Z The above implies that Z the model corresponds to an inventory system in which backlogging of orders is permitted.

_{t}, u_{t}, k} (see Sections 1.1, 1.5 and 1.5), where u_{t}is a stochastic variable, can be regarded as a random walk of the process {Z_{t}}. Suppose a particle traverses a straight line path; at the beginning of each time interval (t, t+1) or (τ_{t}, τ_{t+1}), where t =..., -1,0,1,2,..., the particle is at position Z (as measured from the origin 0), and immediately after has a displacement u (-∞ < u_{t}< ∞); then if it does not have any further displacement, the position of the particle at the end of the time period is given by$${Z_{{t + 1}}} = \min {\{ {Z_{t}} + {u_{t}},k\} ^{ + }}$$

(2.1)

_{t+1}cannot be negative, at the same time if Z_{t}+ u_{t}exceeds k, the particle continues to be at k; thus we have two impenetrable barriers at 0 and k. If k = ∞, then we have a walk on a line of infinite length with a single impenetrable barrier at 0. Similarly we can conceive of a walk Z_{t}which has only one barrier at k but the line runs to infinity in the negative zone; here we get$${Z_{{t + 1}}} = \min \{ {Z_{t}} + {u_{t}},k\} , - \infty < {Z_{t}} \leqslant $$

(2.2)

## Keywords

Service Time Feedback Effect Busy Period Queue Size Duality Relation
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