We study a queueing model with ordered arrivals, which can be called the ∆ (i) /GI/1 queue. Here, customers from a fixed, finite, population independently sample a time to arrive from some given distribution F , and enter the queue in order of the sampled arrival times. Thus, the arrival times are order statistics, and the inter-arrival times are differences of consecutive ordered statistics. They are served by a single server with independent and identically distributed service times, with general service distribution G. The discrete event model is analytically intractable. Thus, we develop fluid and diffusion limits for the performance metrics of the queue. The fluid limit of the queue length is observed to be a reflection of a 'fluid netput' process, while the diffusion limit is observed to be a function of a Brownian motion and a Brownian bridge process or 'diffusion netput' process. The diffusion limit can be seen as being reflected through the directional derivative of the Skorokhod regulator of the fluid netput process in the direction of the diffusion netput process. We also observe what may be interpreted as a sample path Little's law. Sample path analysis reveals various operating regimes where the diffusion limit switches between a free diffusion, a reflected diffusion process and the zero process, with possible discontinuities during regime switches. The weak convergence results are established in the M 1 topology.Keywords Queueing models · transient queueing systems · fluid and diffusion limits · distributional approximations · directional derivatives · M 1 topology
Queueing networks are typically modelled assuming that the arrival process is exogenous, and unaffected by admission control, scheduling policies, etc. In many situations, however, users choose the time of their arrival strategically, taking delay and other metrics into account. In this paper, we develop a framework to study such strategic arrivals into queueing networks. We start by deriving a functional strong law of large numbers (FSLLN) approximation to the queueing network. In the fluid limit derived, we then study the population game wherein users strategically choose when to arrive, and upon arrival which of the K queues to join. The queues start service at given times, which can potentially be different. We characterize the (strategic) arrival process at each of the queues, and the price of anarchy of the ensuing strategic arrival game. We then extend the analysis to multiple populations of users, each with a different cost metric. The equilibrium arrival profile and price of anarchy are derived. Finally, we present the methodology for exact equilibrium analysis. This, however, is tractable for only some simple cases such as two users arriving at a two node queueing network, which we then present.
We introduce the ∆ (i) /GI/1 queue, a new model of transitory queueing, where demand for service exists only in a fixed interval of time and a large, but finite, number of customers enter the system. Customers independently arrive according to some given distribution F . Thus, the arrival times are an ordered statistics, and the inter-arrival times are differences of consecutive ordered statistics. They are served by a single server which provides service according to a general distribution G, with independent service times. The exact model is analytically intractable. Thus, we develop fluid and diffusion limits for the various stochastic processes as the population size is increased. The fluid limit of the queue length is observed to be a reflected process, while the diffusion limit is observed to be a function of a Brownian motion and a Brownian bridge, and given by a netput process and a directional derivative of the Skorokhod reflected fluid netput in the direction of a diffusion refinement of the netput process.
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