Queing Theory, a Tool for Polio Eradication in Nigeria

To curb the spread of contagious diseases and the recent polio outbreak in Nigeria, health departments must set up and operate clinics to dispense medications or vaccines. Residents arrive according to an external (not necessarily Poisson) Arrival process to the clinic. When a resident arrives, he goes to the first workstation, based on his or her information, the resident moves from one workstation to another in the clinic. The queuing network is decomposed by estimating the performance of each workstation using a combination of exact and approximate models. A key contribution of this research is to introduce approximations for workstations with batch arrivals and multiple parallel servers, for workstations with batch service processes and multiple parallel servers, and for self service workstations. We validated the models for likely scenarios using data collected from one of the states vaccination clinics in the country during the vaccination exercises.

Study on Transient Queue-Size Distribution in the Finite-Buffer Model with Batch Arrivals and Multiple Vacation Policy

The transient behavior of the finite-buffer queueing model with batch arrivals and generally distributed repeated vacations is analyzed. Such a system has potential applications in modeling the functioning of production systems, computer and telecommunication networks with energy saving mechanism based on cyclic monitoring the queue state (Internet of Things, wireless sensors networks, etc.). Identifying renewal moments in the evolution of the system and applying continuous total probability law, a system of Volterra-type integral equations for the time-dependent queue-size distribution, conditioned by the initial buffer state, is derived. A compact-form solution for the corresponding system written for Laplace transforms is obtained using an algebraic approach based on Korolyuk’s potential method. An illustrative numerical example presenting the impact of the service rate, arrival rate, initial buffer state and single vacation duration on the queue-size distribution is attached as well.

Simulation of railway marshalling yards using the methods of the queueing theory

Aim. The paper primarily aims to simulate the operation of railway transportation systems using the queueing theory with the case study of marshalling yards. The goals also include the development of the methods and tools of mathematical simulation and queueing theory. Methods. One of the pressing matters of modern science is the development of methods of mathematical simulation of transportation systems for the purpose of analyzing the efficiency, stability and dependability of their operation while taking into account random factors. Research has shown that the use of the most mature class of such models, the singlephase Markovian queueing systems, does not enable an adequate description of transportation facilities and systems, particularly in railway transportation. For that reason, this paper suggests more complex mathematical models in the form of queueing networks, i.e., multiple interconnected queueing systems, where arrivals are serviced. The graph of a queueing network does not have to be connected and circuit-free (a tree), which allows simulating transportation systems with random structures that are specified in table form as a so-called “routing matrix”. We suggest using the BMAP model for the purpose of describing incoming traffic flows. The Branch Markovian Arrival Process is a Poisson process with batch arrivals. It allows combining several different arrivals into a single structure, which, in turn, significantly increases the simulation adequacy. The complex structure of the designed model does not allow studying it analytically. Therefore, based on the mathematical description, a simulation model was developed and implemented in the form of software. Results. The developed models and algorithms were evaluated using the case study of the largest Russian marshalling yard. A computational experiment was performed and produced substantial recommendations. Another important result of the research is that significant progress was made in the development of a single method of mathematical and computer simulation of transportation hubs based on the queueing theory. That is the strategic goal of the conducted research that aims to improve the accuracy and adequacy of simulation compared to the known methods, as well as should allow extending the capabilities and applicability of the model-based approach. Conclusions. The proposed model-based approach proved to be a rather efficient tool that allows studying the operation of railway marshalling yards under various parameters of arrivals and different capacity of the yards. It is unlikely to completely replace the conventional methods of researching the operation of railway stations based on detailed descriptions. However, the study shows that it is quite usable as a primary analysis tool that does not require significant efforts and detailed statistics.

Response Time Distribution in a Tandem Pair of Queues with Batch Processing

Response time density is obtained in a tandem pair of Markovian queues with both batch arrivals and batch departures. The method uses conditional forward and reversed node sojourn times and derives the Laplace transform of the response time probability density function in the case that batch sizes are finite. The result is derived by a generating function method that takes into account that the path is not overtake-free in the sense that the tagged task being tracked is affected by later arrivals at the second queue. A novel aspect of the method is that a vector of generating functions is solved for, rather than a single scalar-valued function, which requires investigation of the singularities of a certain matrix. A recurrence formula is derived to obtain arbitrary moments of response time by differentiation of the Laplace transform at the origin, and these can be computed rapidly by iteration. Numerical results for the first four moments of response time are displayed for some sample networks that have product-form solutions for their equilibrium queue length probabilities, along with the densities themselves by numerical inversion of the Laplace transform. Corresponding approximations are also obtained for (non-product-form) pairs of “raw” batch-queues—with no special arrivals—and validated against regenerative simulation, which indicates good accuracy. The methods are appropriate for modeling bursty internet and cloud traffic and a possible role in energy-saving is considered.

Analysis of Single-Server Multi-Class Queue with Unreliable Service, Batch Correlated Arrivals, Customers Impatience, and Dynamical Change of Priorities

A single-server non-pre-emptive priority queueing system of a finite capacity with many types of customers is analyzed. Inter-arrival times can be correlated and batch arrivals are allowed. Possible unreliability of the server, implying the loss of a customer or the necessity of its service from the early beginning or some phase of the service, is taken into account. Initial priorities provided to various types of customers at the arrival moment can be varied (increased or decreased) after the random amount of time during the customer stay in the buffer. Such a type of queues arises in the modeling operation of various emergency care systems, information, and perishable goods delivering systems, etc. The stationary behavior of the system is described by the finite state multi-dimensional continuous-time Markov chain with the upper-Hessenberg block structure of the generator. The stationary distribution of the system states and some important characteristics of the system are calculated. The presented numerical examples illustrate opportunities to quantitatively evaluate the impact of the buffer capacity and customers’ mean arrival rate on the most important characteristics of the system. The possibility of solving optimization problems is briefly shown.