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Inventory models with lost sales and large lead times have traditionally beenconsidered intractable due to the curse of dimensionality. Recently, Goldbergand co-authors laid the foundations for a new approach to solving these models,by proving that as the lead time grows large, a simple constant-order policy isasymptotically optimal. However, the bounds proven there require the lead timeto be very large before the constant-order policy becomes effective, incontrast to the good numerical performance demonstrated by Zipkin even forsmall lead time values. In this work, we prove that for the infinite-horizonvariant of the same lost sales problem, the optimality gap of the sameconstant-order policy actually converges \emph{exponentially fast} to zero,with the optimality gap decaying to zero at least as fast as the exponentialrate of convergence of the expected waiting time in a related single-serverqueue to its steady-state value. We also derive simple and explicit bounds forthe optimality gap, and demonstrate good numerical performance across a widerange of parameter values for the special case of exponentially distributeddemand. Our main proof technique combines convexity arguments with ideas fromqueueing theory.

Optimality gap of constant-order policies decays exponentially in the lead time for lost sales models