Queue and Stochastic Models for Operations Research II

This paper considers a tandem queueing network with a Poisson arrival process of incoming calls, two servers, and two infinite orbits by the method of asymptotic analysis. The servers provide services for incoming calls for exponentially distributed random times. Blocked customers at each server join the orbit of that server and retry to enter the server again after an exponentially distributed time. Under the condition of low retrial rates, we prove that the joint stationary distribution of scaled numbers of calls in the orbits weakly converges to a two-variable Normal distribution. Full article

Several studies have been conducted on scaling limits for Markov-modulated infinite-server queues. To the best of our knowledge, most of these studies adopt an approach to prove the convergence of the moment-generating function (or characteristic function) of the random variable that represents a scaled version of the number of busy servers and show the weak law of large numbers and the central limit theorem (CLT). In these studies, an essential assumption is the finiteness of the phase process and, in most of them, the CLT for the number of busy servers conditional on the phase (or the joint states) has not been considered. This paper proposes a new method called the moment approach to address these two limitations in an infinite-server batch service queue, which is called the M/M${}^X$/∞ queue. We derive the conditional weak law of large numbers and a recursive formula that suggests the conditional CLT. We derive series expansion of the conditional raw moments, which are used to confirm the conditional CLT by a symbolic algorithm. Full article

We consider a discrete-time single server queueing system, where arrivals stem from a multi-type Galton–Watson branching process with migration. This branching-type arrival process exhibits intricate correlation, and the performance of the corresponding queueing process can be assessed analytically. We find closed-form expressions for various moments of both the queue content and packet delay. Close inspection of the arrival process at hand, however, reveals that sample paths consist of large independent bursts of arrivals followed by geometrically distributed periods without arrivals. Allowing for non-geometric periods without arrivals, and correlated bursts, we apply $\pi$-thinning on the arrival process. As no closed-form expressions can be obtained for the performance of the corresponding queueing system, we focus on approximations of the main performance measures in the light and heavy traffic regimes. Full article

We present the study of a non-classical discrete-time queuing system in which the customers each request a variable amount of service, called their “service demand”, from a system with multiple servers, each of which can provide a variable amount of service, called their “service capacity”, in each time slot. The service demands are independent from customer to customer and follow a general distribution, whereas the service capacities follow a phase-type distribution and are independent from server to server and from slot to slot. Since an exact analytical analysis for this general queuing system is infeasible, we propose several approximations for the key performance characteristics in this system such as the mean system content and the mean customer delay in steady state. The accuracy of each of these approximations is compared to simulations using several numerical examples. Full article

The need for transporting commodities has led to more and more freight vehicles on urban roads. Specific operational constraints of such vehicles could induce non-homogeneities in the smooth movement of traffic, especially at intersections where acceleration/deceleration events occur frequently. This leads to unnecessary wasted time for all vehicles, even in low to moderate traffic conditions. Hence, the literature reports different proposals to enhance the continuity of traffic at intersections. Among them, the green extension strategy has attracted researchers’ attention, owing to its simplicity, flexibility and practicality. In this paper, we propose a new approximate probabilistic model for the expected waiting/wasted time of all vehicles at an intersection with green time extension in low to moderate traffic conditions. Accordingly, the optimal green extension interval that minimizes the total expected waiting time can then be determined in different conditions. The proposed analysis needs few pieces of information (as opposed to microsimulation models conventionally employed to analyze such systems) and is therefore, suitable for quickly deciding on the optimal strategy based on the current situation in a dynamic environment. We have validated our approximate analysis with simulations in the VISSIM simulation tool. Full article

In this research, a single-server M-class retrial queueing system (orbit queue) with constant retrial rates and Poisson inputs is considered. The main purpose is to construct the upper and lower bounds of the stationary workload in this system expressed via the stationary workloads in the classical $M/G/1$ systems where the service time has M-component mixture distributions. This analysis is applied to establish the extreme behaviour of stationary workload in the retrial system with Pareto service-time distributions for all classes. Full article

The Asymmetric Simple Inclusion Process (ASIP) is an n-site tandem stochastic network with a Poisson arrival influx into the first site. Each site has an unlimited buffer with a gate in front of it. Each gate opens, independently of all other gates, following a site-dependent Exponential inter-opening time. When a site’s gate opens, all particles occupying the site move simultaneously to the next site. In this paper, a Generalized ASIP network is analyzed where the influx is to all sites, while gate openings are determined by a general renewal process. A compact matrix approach—instead of the conventional (and tedious) successive substitution method—is constructed for the derivation of the multidimensional probability-generating function (PGF) of the site occupancies. It is shown that the set of $\left(\begin{array}{c}2n\\ n\end{array}\right)$ linear equations required to obtain the PGF of an n-site network can be first cut by half into a set of $\left(\begin{array}{c}2n-1\\ n\end{array}\right)$ equations, and then further reduced to a set of $2^n-\left(n+1\right)$ equations. The latter set can be additionally split into several smaller triangular subsets. It is also shown how the PGF of an $\left(n+1\right)$-site network can be derived from the corresponding PGF of an n-site system. Explicit results for networks with $n=3$ and $n=4$ sites are obtained. The matrix approach is utilized to explicitly calculate the probability that site $k \left(k=1,2,\dots ,n\right)$ is occupied. We show that, in the case where arrivals occur to the first site only, these probabilities are functions of both the site’s index and the arrival flux and not solely of the site’s index. Consequently, refined formulas for the latter probabilities and for the mean conditional site occupancies are derived. We further show that in the case where the arrival process to the first site is Poisson with rate $\lambda$, the following interesting property holds: $P\left(sitekisoccupied|\lambda =1\right)=P\left(sitek+1isoccupied|\lambda \to \infty \right)$. The case where the inter-gate opening intervals are Gamma distributed is investigated and explicit formulas are obtained. Mean site occupancy and mean total load of the first $k$ sites are calculated. Numerical results are presented. Full article

This research contributes to the security design of an advanced smart drone swarm network based on a variant of the Blockchain Governance Game (BGG), which is the theoretical game model to predict the moments of security actions before attacks, and the Strategic Alliance for Blockchain Governance Game (SABGG), which is one of the BGG variants which has been adapted to construct the best strategies to take preliminary actions based on strategic alliance for protecting smart drones in a blockchain-based swarm network. Smart drones are artificial intelligence (AI)-enabled drones which are capable of being operated autonomously without having any command center. Analytically tractable solutions from the SABGG allow us to estimate the moments of taking preliminary actions by delivering the optimal accountability of drones for preventing attacks. This advanced secured swarm network within AI-enabled drones is designed by adapting the SABGG model. This research helps users to develop a new network-architecture-level security of a smart drone swarm which is based on a decentralized network. Full article