In the Operations Research queue-joining literature, beginning with Naor (1969), the queue joining probability is monotonic decreasing in the queue length; the longer the queue, the fewer consumers join. Recent academic and empirical evidence indicates that queue-joining probabilities may not always be decreasing in the queue length. We provide a simple explanation for these non-monotonic queue...

Queues with Markovian arrival and service processes, i.e., MAP/MAP/1 queues, have been useful in the analysis of computer and communication systems and di erent representations for their stationary sojourn time and queue length distribution have been derived. More speci cally, the class of MAP/MAP/1 queues lies at the intersection of the class of QBD queues and the class of semi-Markovian queue...

Inspired by the need for efficient traffic control in urban areas, we study the problem of minimizing the longest queue length in traffic networks. In our model cars only move forward and do not overtake each other and time is divided in rounds of a given length. Cars always cross intersections when they have the green signal and they are at the front of the queue, otherwise they wait. Each car...

We study the impact of service-time distributions on the distribution of the maximum queue length during a busy period for the M/G/1 queue. The maximum queue length is an important random variable to understand when designing the buffer size for finite buffer (M/G/1/n) systems. We show the somewhat surprising result that for three variations of the preemptive LCFS discipline, the maximum queue ...

We consider the FCFS GI/GI/n queue in the Halfin-Whitt heavy traffic regime, in the presence of heavy-tailed distributions (i.e. infinite variance). We prove that under minimal assumptions, i.e. only that processing times have finite 1 + moment for some > 0 and inter-arrival times have finite second moment, the sequence of stationary queue length distributions, normalized by n 1 2 , is tight in...

We present a heavy traffic analysis for a single server queue with renewal arrivals and generally distributed i.i.d. service times, in which the server employs the Shortest Remaining Processing Time (SRPT) policy. Under typical heavy traffic assumptions, we prove a diffusion limit theorem for a measure-valued state descriptor, from which we conclude a similar theorem for the queue length proces...

The problem considered is that of estimating the tail stationary probability for two exponential server queues in series fed by renewal arrivals. We compute the tail of the marginal queue length distribution at the second queue. The marginal at the first queue is known by the classical result for the GI/M/1 queue. The approach involves deriving necessary and sufficient conditions on the paths o...

We study the steady-state queue-length vector in a multi-class queue with relative priorities. Upon service completion, the probability that the next served customer is from class k is controlled by class-dependent weights. Once a customer has started service, it is served without interruption until completion. We establish a state-space collapse for the scaled queue length vector in the heavy-...