Two-phase Queueing System Research Articles

New Analytic Solutions of Queueing System for Shared–Short Lanes at Unsignalized Intersections

Designing the crossroads capacity is a prerequisite for achieving a high level of service with the same sustainability in stochastic traffic flow. Also, modeling of crossroad capacity can influence on balancing (symmetry) of traffic flow. Loss of priority in a left turn and optimal dimensioning of shared-short line is one of the permanent problems at intersections. A shared–short lane for taking a left turn from a priority direction at unsignalized intersections with a homogenous traffic flow and heterogeneous demands is a two-phase queueing system requiring a first in–first out (FIFO) service discipline and single-server service facility. The first phase (short lane) of the system is the queueing system M(pλ)/M(μ)/1/∞, whereas the second phase (shared lane) is a system with a binomial distribution service. In this research, we explicitly derive the probability of the state of a queueing system with a short lane of a finite capacity for taking a left turn and shared lane of infinite capacity. The presented formulas are under the presumption that the system is Markovian, i.e., the vehicle arrivals in both the minor and major streams are distributed according to the Poisson law, and that the service of the vehicles is exponentially distributed. Complex recursive operations in the two-phase queueing system are explained and solved in manuscript.

Optimal control of M/M/1 two-phase queueing system with state-dependent arrival rate, server breakdowns, delayed repair, and N-policy

In this paper, we examine a two-stage queueing system where the arrivals are Poisson with rate depends on the condition of the server to be specific: vacation, pre-service, operational or breakdown state. The service station is liable to breakdowns and deferral in repair because of non-accessibility of the repair facility. The service is in two basic stages, the first being bulk service to every one of the customers holding up on the line and the second stage is individual to each of them. The server works under N-policy. The server needs preliminary time (startup time) to begin batch service after a vacation period. Startup times, uninterrupted service times, the length of each vacation period, delay times and service times follows an exponential distribution. The closed form of expressions for the mean system size at different conditions of the server is determined. Numerical investigations are directed to concentrate the impact of the system parameters on the ideal limit N and the minimum base expected unit cost.

Two-phase queueing system optimization in applications to data transmission control

We consider the controlled tandem queuing network with two single-server finite-length queues with non-stationary Poisson input flow of packets. The first station controls the service rate and drops any incoming packet if its buffer is full. The second station controls the packet acceptance probability. The tandem is described by a controlled finite-state Markov process and optimized on a fixed time interval by minimizing average number of dropped packets with constraints on the average sojourn time and energy consumption.

Two-phase queueing system with unreliable servers under heavy load

We consider a two-phase queueing system with unreliable servers and a finite number of places in a buffer between phases. The servers work synchronously at phases. The ergodicity condition is obtained and the work of the system is studied under heavy traffic assumptions.

Help desk center operating model as a two-phase queueing system

We consider a two-phase queueing system with a Markovian arrival flow as an operating model for a help desk center. The first phase is a multiserver system with a finite buffer and impatient customers. After getting service in the first phase, a customer either enters the second phase with an infinite buffer or quits the system. Service times at the first and second stages have phase-type distributions with different parameters. We obtain an existence condition for a stationary regime of the system. An algorithm for computing stationary probabilities and basic performance characteristics of the system is presented. Laplace-Stieltjes transforms for the distributions of sojourn and waiting times in the first and second phases are found. Results of numerical experiments are presented. Optimization problem for the system operation is solved numerically.

A two-phase GI/PH/1 → ·/PH/1/0 system with losses

We find a stationary distribution and productivity characteristics for a two-phase queueing system with recurrent input flow. The servicing units are single-line on both phases. The servicing times have phase-type distributions. On the first phase, there is an infinite buffer. There is no intermediate buffer. In case the second phase device is busy at the time a claim has finished going through the first phase, this claim leaves the system unserved (is lost).

Analysis of a two-phase queueing system with a fixed-size batch policy

We consider a single-server, two-phase queueing system with a fixed-size batch policy. Customers arrive at the system according to a Poisson process and receive batch service in the first-phase followed by individual services in the second-phase. The batch service in the first-phase is applied for a fixed number ( k) of customers. If the number of customers waiting for the first-phase service is less than k when the server completes individual services, the system stays idle until the queue length reaches k. We derive the steady state distribution for the system’s queue length. We also show that the stochastic decomposition property can be applied to our model. Finally, we illustrate the process of finding the optimal batch size that minimizes the long-run average cost under a linear cost structure.

A two-phase queueing system with repeated attempts and Bernoulli vacation schedule

This paper deals with the steady-state behaviour of an M/G/1 queue with repeated attempts in which the server provides two phases of heterogeneous service under Bernoulli vacation schedule. We carry out an extensive analysis of the system, including existence of stationary regime, embedded Markov chain, steady-state distribution of the server state and the number of customer in the orbit and some system performance measures. This model generalises both the classical M/G/1 retrial queue and the M/G/1 queue with two phases of service and Bernoulli vacation model.

Sojourn time analysis of a queueing system with two‐phase service and server vacations

AbstractWe consider a two‐phase service queueing system with batch Poisson arrivals and server vacations denoted by MX/G1‐G2/1. The first phase service is an exhaustive or a gated bulk service, and the second phase is given individually to the members of a batch. By a reduction to an MX/G/1 vacation system and applying the level‐crossing method to a workload process with two types of vacations, we obtain the Laplace–Stieltjes transform of the sojourn time distribution in the MX/G1‐G2/1 with single or multiple vacations. The decomposition expression is derived for the Laplace–Stieltjes transform of the sojourn time distribution, and the first two moments of the sojourn time are provided. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007

Two-Phase Queueing System with a Markov Arrival Process and Blocking

Two-Phase Queueing System with a Markov Arrival Process and Blocking

A Two-Phase BMAP|G|1|N → PH|1|M – 1 System with Blocking

The stationary probability distribution of a two-phase queueing system with a finite or an infinite buffer for the first phase and a finite buffer for the second phase is derived. The input flow of the system is a batch Markov arrival process. Both phases have single-servers. The service time distribution is arbitrary for the first phase and of phase-type for the second phase. If the buffer of the second phase is full at the instant of completion of service at the first phase, the first server is blocked until the buffer is freed.

Sojourn time analysis of a two-phase queueing system with exhaustive batch-service and its vacation model

We consider an M/G/1-type, two-phase queueing system, in which the two phases in series are attended alternatively and exhaustively by a moving single-server according to a batch-service in the first phase and an individual service in the second phase. We show that the two-phase queueing system reduces to a new type of single-vacation model with non-exhaustive service. Using a double transform for the joint distribution of the queue length in each phase and the remaining service time, we derive Laplace-Stieltjes transforms for the sojourn time in each phase and the total sojourn time in the system. Furthermore, we provide the moment formula of sojourn times and numerical examples of an approximate density function of the total sojourn time.

Performance Analysis of a Discrete-Time Two-Phase Queueing System

This paper introduces the modeling and analysis of a discrete-time, two-phase queueing system for both exhaustive batch service and gated batch service. Packets arrive at the system according to a Bernoulli process and receive batch service in the first phase and individual services in the second phase. We derive the probability generating function (PGF) of the system size and show that it is decomposed into two PGFs, one of which is the PGF of the system size in the standard discrete-time Geo/G/1 queue without vacations. We also present the PGF of the sojourn time. Based on these PGFs, we present useful performance measures, such as the mean number of packets in the system and the mean sojourn time of a packet.

Cycle analysis of a two-phase queueing model with threshold

We consider a single-server, two-phase queueing system with N-policy. Customers arrive at the system according to a Poisson process and receive batch service in the first phase followed by individual services in the second phase. If the system becomes empty at the moment of the completion of the second-phase services, it is turned off. After an idle period, when the queue length reaches N (threshold), the server is turned on and begins to serve customers. We obtain the system size distribution and show that the system size decomposes into three random variables. The system sojourn time is provided. Analysis for the gated batch service model is also provided. Finally we derive a condition under which the optimal operating policy is achieved.

A two-phase queueing system with server vacations

We consider a two-phase queueing system with server vacations. Customers receive a batch service in the first phase and individual services in the second phase. The server takes vacations as soon as the system becomes empty. The model is analyzed using the Laplace-Stieltjes transform. This extends the model studied in Doshi [3].