Stochastic 1D search-and-capture as a <i>G/M/c</i> queueing model

Abstract We study the accumulation of resources within a target due to the interplay between continual delivery, driven by 1D stochastic search processes, and sequential consumption. The assumption of sequential consumption is key because it changes the commonly used <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> queue to a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:math> queue. Combining the theory of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:math> queues with the theory of first-passage times, we derive general conditions for the search process to ensure that the number of resources within the queue converges to a steady state and compute explicit expressions for the mean and variance of the number of resources within the queue at steady state. We then compare the performance of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:math> queue with that of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> queue for an increasing number of servers. We extend the model to consider two competing targets and show that, under specific scenarios, an additional target is beneficial to the original target. Finally, we study the effects of multiple searchers. Using renewal theory, we numerically compute the inter-arrival time density for M searchers in the Laplace space, which allows us to exploit the explicit expressions for the steady-state statistics of the number of resources within <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> queues, and compare their behaviour with different numbers of searchers. Overall, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:math> queue shows a tighter dependence on the configuration of the search process than the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> queue does.