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This paper proposes a totally asymmetric simple exclusion process on atraveling lane, which is equipped with a queueing system and functions of siteassignments along the parking lane. In the proposed system, new particlesarrive at the rear of the queue existing at the leftmost site of the system. Aparticle at the head of the queue selects one of the empty sites in the parkinglane and reserves it for stopping at once during its travel. The arriving timeand staying time in the parking sites follow half-normal distributions. Therandom selections of empty sites are controlled by the bias of the exponentialfunction. Our simulation results show the unique shape of site usagedistributions. In addition, the number of reserved sites is found to increasewith an S-shape curve as the bias to the rightmost site increases. To describethis phenomena, we propose an approximation model, which is derived from thebirth-death process and extreme order statistics. A queueing model that takesthe effect of distance from the leftmost site of the traveling lane intoconsideration is further proposed. Our approximation model properly describesthe distributions of site usage, and the proposed queueing model shows goodagreement with the simulation results.

Effect of walking-distance on a queuing system of totally asymmetric simple exclusion process equipped with functions of site assignments