In our daily life, we often experience waiting in a queue to receive some kind of service. Some customers do not join the queue at the end like other normal customers, and try to cut in the queue hoping to have a shorter waiting time and a higher level of satisfaction. This behaviour is called customer interjection. Some of these customers only try to cut in queue, while some others try to find excuses for interjection. For instance, the first-come-first-served (FCFS) service discipline is usually assumed in public places like restaurants, banks, airports, and supermarkets. However, customer interjections can still be seen in these places. In telecommunications networks, to test the efficiency of transmission, artificial packages are inserted into the normal traffic in a random manner. These interjections can affect the waiting time of other customers in queue. Such interjections may reduce the waiting time of interjecting customers, but increase the waiting time and dissatisfaction of others. In this work, an M/M/1 queueing system with customer interjection is investigated. The arrival of customers to the system is assumed to be a Poisson process with arrival rate . The service times for customers are independent and identically distributed random variables with an exponential distribution with rate . Customers are dispersed into normal customers and interjecting customers. A normal customer joins the queue at the end, and an interjecting customer tries to cut in the queue and occupy a position as close to the head of the queue as possible. Two parameters are introduced to describe the interjection behaviour: the percentage of customers interjecting and the tolerance level of interjection by individual customers who are already waiting in the queue. Using matrix-analytic methods and stochastic comparison methods, the waiting times of normal customers and interjecting customers are being studied. The impacts of the two parameters on the waiting times are analyzed in detail, and the implications of the results are discussed with numerical examples. It is found that the waiting times are sensitive to the tolerance level of interjection by individual customers. It is also found that eliminating customer interjection would be always beneficial to normal customers and arbitrary customers though it would not always be so for interjecting customers.

In this thesis, we study the tradeoff of average delay with average service cost and average utility for both continuous time and discrete time single server queueing models without and with admission control. The continuous time and discrete time queueing models that we consider are motivated by cross-layer models for point-to-point links with random packet arrivals and fading at slow and fast time scales. Our studies are motivated by the need to optimally tradeoff the average delay of the packets (a network layer performance measure) with the average service cost of transmitting the packets, e.g. the average power required for transmission (a physical layer performance measure) under a lower bound constraint on the average throughput, in various point-to-point communication scenarios. The tradeoff problems are studied for a class of monotone and stationary scheduling policies and under the assumption that the service cost rate and utility rate are respectively convex and concave functions of the service rate and arrival rate. We also consider the problem of optimally trading off the average delay and average error rate of randomly arriving message symbols which are transmitted over a noisy point-to-point link, in which case the service cost function is non-convex. The solutions to the tradeoff problems that we address in the thesis are asymptotic in nature, and are similar in spirit to the Berry-Gallager asymptotic bounds. It is intuitive that to keep a queue stable under a lower bound constraint on the average utility a minimum number of customers have to be served per unit time. This in turn implies that queue stability requires a minimum average service cost expenditure. In the thesis we obtain an asymptotic characterization of the minimum average delay for monotone stationary policies subject to an upper bound constraint on the average service cost and a lower bound constraint on the average utility, in the asymptotic regime where the average service cost constraint is made arbitrarily close to the above minimum average service cost. In the thesis, we obtain asymptotic lower bounds on the minimum average delay for the cases for which lower bounds were previously not known. The asymptotic characterization of the minimum average delay for monotone stationary policies, for both continuous time and discrete time models, is obtained via geometric bounds on the stationary probability of the queue length, in the above asymptotic regime. The restriction to monotone stationary policies enables us to obtain an intuitive explanation for the behaviour of the asymptotic lower bounds using the above geometric bounds on the stationary probability distribution of the queue length. The geometric bounds on the stationary probability of the queue length also lead to a partial asymptotic characterization of the structure of any optimal monotone stationary policy, in the above asymptotic regime, which was not available in previous work. Furthermore, the geometric bounds on the stationary probability can be extended to analyse the tradeoff problem in other scenarios, such as for other continuous time queueing models, multiple user communication models, queueing models with service time control, and queueing models with general holding costs. Usually, queueing models with integer valued queue evolution, are approximated by queueing models with real valued queue evolution and strictly convex service cost functions for analytical tractability. Using the asymptotic bounds, we show that for some cases the average delay does not grow to infinity in the asymptotic regime, although the approximate model suggests that the average delay does grow to infinity. In other cases where the average delay does grow to infinity in the asymptotic regime, our results illustrate that the tradeoff behaviour of the approximate model is different from that of the original integer valued queueing model unless the service cost function is modelled as the piecewise linear lower convex envelope of the service cost function for the original model.

Firm real time system with Poisson arrival process, iid exponential service times and iid deadlines till the end of service of a job, operated under the First Come First Served (FCFS) scheduling policy is well studied. In this thesis, we present an exact theoretical analysis of a similar (M/M/1 + G queue) system with exact admission control (EAC). We provide an explicit expression for the steady state workload distribution. We use this solution to derive explicit expressions for the loss ratio and the sojourn time distribution. An exact theoretical analysis of the performance of an M/M/1 + G queue with preemptive deadlines till the end of service, operating under the Earliest Deadline First (EDF) scheduling policy, appears to be difficult, and only approximate formulas for the loss ratio are available in the literature. We present in this thesis similar approximate formulas for the loss ratio in the present of an exit control mechanism, which discards a job at the epoch of its getting the server if there is no chance of completing it. We refer to this exit control mechanism as the Early job Discarding Technique (EDT). Monte Carlo simulations of performance indicate that the maximum approximation error is reasonably small for a wide range of arrival rates and mean deadlines. Finally, we compare the loss ratios of the First Come First Served and the Earliest Deadline First scheduling policies with or without admission or exit control mechanism, as well as their counterparts with deterministic deadlines. The results include some formal equalities, inequalities and some counter-examples to establish non-existence of an order. A few relations involving loss ratios are posed as conjectures, and simulation results in support of these are reported. These results lead to a complete picture of dominance and non-dominance relations between pairs of scheduling policies, in terms of loss ratios.