Finite Capacity Queueing System Research Articles

Two-Level Full Factorial Design Approach for the Analysis of Multi-Lane Highway Section under Saturated and Unsaturated Traffic Flow Conditions

Oversaturation of highways occurs due to their inadequate assessment and design. In this paper, we propose both a mathematical queuing model and a Discrete-Event Simulation (DES) framework based on Newell’s triangular flow-density relationship for the performance analysis of a multi-lane highway section. The proposed framework is a finite capacity queuing system, which captures an increase in the flow with the vehicle density up to the capacity of the section in an unsaturated condition and a decrease in the flow in the case of a saturated condition, depicting the actual traffic conditions on the highway section. First, the Birth–Death Process is used to build the mathematical queuing model (BDP), and the average number of vehicles (average queue length) and blocking probability on the highway section are estimated. Then, the accuracy of the mathematical queuing model is verified by the proposed DES framework. The “significance and effects” of different design factors are evaluated using the two-level full factorial design technique. The analysis of the experimental results reveals that the length of the highway section and the number of lanes are the most significant factors affecting the average queue length and blocking probability, while the jam density only has a significant effect on the average queue length and does not affect the blocking probability. In case of a two-way interaction, the combined effect of the “length-lanes” significantly affects the average queue length. In the end, a multiple-factor linear regression model is also developed for the prediction of the average number of vehicles on the highway section based on the design factors.

A meta-heuristic optimization for a novel mathematical model for minimizing costs and maximizing donor satisfaction in blood supply chains with finite capacity queueing systems

Blood and blood banks play an essential role in patient care. This study proposes a novel mathematical model for Blood Supply Chains (BSCs), which minimizes the total costs and maximizes donor satisfaction by reducing the system’s waiting time. The model addresses several aspects of the BSC, such as locating collection centers, allocating blood donors to collection centers, inventory control in the main blood center, blood shortage and deterioration considerations, and queueing systems in collection centers over a planning horizon. Queueing systems lead to a better understanding of the BSC and provide an opportunity to investigate donors’ attendance and waiting times. A real-world case study is presented and solved using a new meta-heuristic algorithm to illustrate the model’s applicability. Finally, sensitivity analyses are performed, and several beneficial managerial insights are provided.

Modified State-Dependent Queuing Model for the Capacity Analysis of Metro Rail Transit Station Corridor during COVID-19

The COVID-19 pandemic policies have had a significant impact on the daily commuter flow at the metro rail transit stations. In this study, we propose a modified state-dependent M(n)/G(n)/C/C queuing model for the analysis of commuter flow in the corridor of metro rail transit stations in the COVID-19 situation in order to ensure safe social distance. The proposed model is a finite capacity queuing system with state-dependent commuter arrivals and state-dependent service rates based on the flow–density relationship. First, a mathematical queuing model is developed by using the birth–death process (BDP) and the expected number of commuters, and average area occupied per commuter and blocking probabilities are computed. Then, the accuracy of the proposed model is verified by a discrete-event simulation (DES) framework. (1) The proposed model’s results are compared to those of the existing M/G(n)/C/C model. The proposed modified model’s sensitivity analysis revealed that the anticipated number of commuters in the corridor remains smaller when the arrival rate is state-dependent. (2) In accordance with COVID-19 protocol, when the facility is congested, commuters are discouraged from entering and a safe social distance is maintained between them. (3) No commuters are impeded, and adequate throughput is ensured from the corridor. The proposed model will assist the metro rail transit station operators in making intelligent decisions regarding the operations in the COVID-19 situation.

Throughput Analysis of an Amazon Go Retail under the COVID-19-related Capacity Constraints

The throughput of a finite-capacity queueing system is the mean number of clients served during a time interval. The COVID-19 outbreak has posed a serious challenge for many commercial establishments, including the retails, which have struggled to adapt to new working dynamics. Retails have been forced to adjust their service guidelines to comply with biosecurity protocols, ensuring to observe governmental and public health policies. A significant change for the retail market has been the capacity restrictions to ensure social distancing, i.e., a limitation on the number of customers simultaneously shopping in the store. Such a constraint has an impact on the throughput that can be achieved by a retail. This article assesses the impact of the capacity restriction measures on an Amazon Go-like retail performance through a throughput analysis under COVID-19-related capacity restrictions. For the assessment, we first retrieved real data from a retail located in Cartagena, Colombia. Two scenarios were considered: i) low demand and ii) high demand. Further, we built an Amazon Go-like, two-queue, M/M/c/K retail model with a CONWIP (Constant Work-In-Process) approach, considering biosecurity-based capacity restrictions due to the COVID-19 pandemic. The R package ‘queueing’ was used to set up the model, and an algorithm was created to go over each sampling period and find the hourly optimum capacity and throughput under the dynamic conditions of both scenarios (low and high demand). Results from the performance analysis show that, for some operational conditions, the optimum maximum throughput is achieved with capacities below the biosecurity-based capacity, while for some other operational conditions the maximum throughput cannot be achieved with the restrictions, as the optimum capacity lies beyond the biosecurity-based capacity. These results suggest that the maximum capacity definition should not be static. Instead, it should be done considering the retail’s dimensions, the biosecurity policies, and the dynamic retail’s operational conditions such as the demand and service capacity.

A queueing model for a wireless sensor node using energy harvesting

In this paper we propose a generic queueing model that can be used to evaluate the performance of a wireless sensor node that uses energy harvesting. The alteration of such a device between the transmit and sleep mode (or between consuming energy and harvesting energy), is modeled by means of a finite capacity queueing system with repeated server vacations. The duration of a service, resp. vacation, is determined by the available energy at the start of the service, resp. vacation. Therefor we introduce in the model a variable that keeps track of the available energy. The system occupancy and the available energy are observed at inspection instants (i.e., the end of a service or of a vacation), resulting in a discrete-time Markov Chain. We derive closed form formulas for the system occupancy distribution at inspection instants and at arbitrary time instants together with the Laplace transform of the waiting time distribution. The possible use of the model to evaluate the system’s performance for various parameter values is illustrated by means of a number of examples.

Optimal and Sensitivity Analysis of Vacation Queueing System with F-Policy and Vacation Interruption

In this article, the investigation on a randomized arrival control policy for the prospective customers in the finite capacity queueing system with working vacation and vacation interruption is done. The impatience behavior of the customers is also considered in modeling and assumptions to the studied problem to make it more realistic. In the investigated queueing model, at the epoch when the number of customers in the system reaches system’s capacity, newly arriving customers are not allowed to join the system for service and referred as lost customers. As the length of the queue decreases to a pre-specified threshold value F, the server commences a start-up for allowing to join the customers according to an exponential distribution and starts allowing newly arriving customers to join the system for service. The steady-state probability distribution and vector representation of various system performance measures are derived using matrix-analytic approach. The cost optimization problem is also formulated, and the particle swarm optimization algorithm is implemented to determine the optimal decision parameters to achieve the minimal expected cost. Finally, some numerical results in tables and graphs are provided for the illustrative and comparative purpose which help the system analyst in decision making from the performance and economic perspective.

Matrix-geometric solution of multi-server queueing systems with Bernoulli scheduled modified vacation and retention of reneged customers: A meta-heuristic approach

ABSTRACT In the present article, we deal with the multi-server finite capacity queueing system with Bernoulli’s scheduled modified vacation policy and the realistic retaining policy of reneged customers. After completion of the service of any customer, the server decides whether to go for the vacation of random duration or to continue facilitating the service to the waiting customer, if any, present in the queue. All servers, homogeneous in nature, provide the state-dependent service following the threshold policy to reduce the overload of the system. The impatience behavior of customers like balking and reneging is also considered in the stochastic modeling of the studied problem. The matrix analytic approach is employed to obtain the steady-state probabilities with which various system performance measures are also developed with practical justification. Finally, the expected cost minimization problem is formulated and dealt with the meta-heuristic approach: particle swarm optimization (PSO). All results of numerical simulation and optimal analysis are summarized in tables and graphs to provide quick insight. The concluding remarks and future scopes have also been discussed.

A Markovian Model for Internet of Things Application

Internet of Things (IoT) allows communication among human-to-things, things-to-human,and things-tothings that are incorporated into an information networks allowing automatic information interchange and the processing of data at real time. In this paper, we conducta performance analysis of a real application defined through four traffic classes withthe priorities present in smart cities using Continuous Time Markov Chains(CTMC). Based on a finite capacity queuing system, we propose a new cost-effective analytical model with a push-out management scheme in favor of the highest priority (emergency) traffic. Based on the analytical model, several performance measures for different traffic classes have been studiedextensively including blocking probability; push out probability, delay, channel utilization as well as overall system performance.

Performance analysis of container yard based on batch service queueing system

China Railway Express has been developed rapidly since “the Belt and Road Initiative” in China was proposed. Many cities in the Midwest of China have developed into the transport hub from the transport terminal. The key question is how to reasonably formulate the size of the container station and the frequency of the shift for China Railway Express. This paper takes the container as the research object. Regarding the containers as customers in the queuing system, customers’ services ending with the loading of containers into China Railway Express, based on a finite-capacity queueing system with batch service, this paper provides an optimal layout method for the container yard of China Railway Express.

Predictor Analysis of the Non-parametric Bulk Arrival Fuzzy Queueing System

In general, queueing methodology is most helpful for design the system and that may achieve the described performance level. This paper, we discuss the fuzzy queueing model with fuzzy parameter. First we construct the membership function of the fuzzy queueing character where the arrival and service rates are triangular fuzzy numbers. Consider the service node as k-phase and to provide the equal service rate in all the phases. Second we shows that the method for constructing the membership function of finite capacity queueing system. A pair of nonlinear program is developed to describe the family of crisp membership functions of finite capacity through which the membership functions of the system performance measures are derived. Finally, we obtain the lower and upper bound of the system performance measure at the different possibility level of alpha. Third we analyze the optimal level of the queueing system, this work extended in [13, 14]. A numerical example is solved successfully.

Simulation of Single-Server Queuing System with Reverse Balking

Jain et. al. (2014) proposed a single-server finite capacity queuing system with reverse balking. The models finds its application in all fields of business, civil, and defence. They perform theoretical analysis of the model. In this paper, we simulate this model. An algorithm is written in ‘C’ language for the same. Theoretical results of model are compared with simulated results numerically. Results are of immense use for understanding gap between theoretical and practical results of the model.

A modified Conway–Maxwell–Poisson type binomial distribution and its applications

ABSTRACTThis article proposes a generalized binomial distribution, which is derived from the finite capacity queueing system with state-dependent service and arrival rates. This distribution is also generated from the conditional Conway–Maxwell–Poisson (CMP) distribution given a sum of two CMP variables. In this article, we consider the properties of the probability mass function, indices of dispersion, skewness and kurtosis, and give applications of the proposed distribution. The estimation method and simulation study are also considered.

An Approximation Method for Blocking Probabilities in M/D/1/K1→ ⋅/D/1/K2Queues

Obtaining exact blocking probabilities for tandem queues with finite capacities is not a trivial problem. In this paper, we propose a computational approximation method using max-plus algebra for computing blocking probability in a Poisson-driven 2-node tandem queue with finite capacities and constant service times. The blocking probability of a finite-capacity queueing system can be obtained from either the tail probability of stationary waiting time or the difference between two expected stationary waiting times at the first node of the corresponding extended 3-node tandem queue. The computational results in this study show that the proposed approach provides a good approximation of the blocking probability, and in particular, it works well under moderately to heavily loaded situations. The proposed approach is not limited to a particular blocking policy, system structure, or service time; hence, it is applicable to general queues with finite buffer capacities and various blocking policies.

Determining optimal treatment rate after a disaster

From the standpoint of medical services, a disaster is a calamitous event resulting in an unexpected number of casualties that exceeds the therapeutic capacities of medical services. In these situations, effective medical response plays a crucial role in saving life. To model medical rescue activities, a two-priority non-preemptive S-server, and a finite capacity queueing system is considered. After constructing Chapman–Kolmogorov differential equations, Pontryagin's minimum principle is used to calculate optimal treatment rates for each priority class. The performance criterion is to minimize both the expected value of the square of the difference between the number of servers and the number of patients in the system, and also the cost of serving these patients over a determined time period. The performance criterion also includes a final time cost related to deviations from the determined value of the desired queue length. The two point boundary value problem is numerically solved for different arrival rate patterns and selected parameters.

Duty-cycle optimization for IEEE 802.15.4 wireless sensor networks

Most applications of wireless sensor networks require reliable and timely data communication with maximum possible network lifetime under low traffic regime. These requirements are very critical especially for the stability of wireless sensor and actuator networks. Designing a protocol that satisfies these requirements in a network consisting of sensor nodes with traffic pattern and location varying over time and space is a challenging task. We propose an adaptive optimal duty-cycle algorithm running on top of the IEEE 802.15.4 medium access control to minimize power consumption while meeting the reliability and delay requirements. Such a problem is complicated because simple and accurate models of the effects of the duty cycle on reliability, delay, and power consumption are not available. Moreover, the scarce computational resources of the devices and the lack of prior information about the topology make it impossible to compute the optimal parameters of the protocols. Based on an experimental implementation, we propose simple experimental models to expose the dependency of reliability, delay, and power consumption on the duty cycle at the node and validate it through extensive experiments. The coefficients of the experimental-based models can be easily computed on existing IEEE 802.15.4 hardware platforms by introducing a learning phase without any explicit information about data traffic, network topology, and medium access control parameters. The experimental-based model is then used to derive a distributed adaptive algorithm for minimizing the power consumption while meeting the reliability and delay requirements in the packet transmission. The algorithm is easily implementable on top of the IEEE 802.15.4 medium access control without any modifications of the protocol. An experimental implementation of the distributed adaptive algorithm on a test bed with off-the-shelf wireless sensor devices is presented. The experimental performance of the algorithms is compared to the existing solutions from the literature. The experimental results show that the experimental-based model is accurate and that the proposed adaptive algorithm attains the optimal value of the duty cycle, maximizing the lifetime of the network while meeting the reliability and delay constraints under both stationary and transient conditions. Specifically, even if the number of devices and their traffic configuration change sharply, the proposed adaptive algorithm allows the network to operate close to its optimal value. Furthermore, for Poisson arrivals, the duty-cycle protocol is modeled as a finite capacity queuing system in a star network. This simple analytical model provides insights into the performance metrics, including the reliability, average delay, and average power consumption of the duty-cycle protocol.

Economic application in a Bernoulli F-policy queueing system with server breakdown

For maintaining the service quality of working place, some kinds of arriving customers control policy are needed. This paper investigates a randomised arriving control policy of finite capacity queuing system with an unreliable server. In the investigated system, when the number of customers reaches the system’s capacity, further arriving customers are not allowed to enter the system. As the queue length decreases to a predefined threshold value, customers may either be allowed entering the system with an exponential startup time or the customers are still unable to enter the system. For such a system, we develop the steady-state solutions and some important system characteristics, such as the probability that the server is busy, effective arrival rate, expected waiting time in the system, etc. We also develop a cost model to determine the optimal control policy at a minimum cost. A sensitivity analysis is also conducted to investigate the effects of changes in the system parameters on the expected cost function. Finally, we present some managerial insights through an application example of the printed circuit boards manufacturing system.

Analysis of GI/M/n/n queueing system with ordered entry and no waiting line

This paper deals with a multi-server, finite-capacity queuing system with recurrent input and no waiting line. The interarrival times are arbitrarily distributed whereas service times are exponentially distributed. Moreover, the servers are heterogeneous and independent of each other. Arriving customers choose the server with the lowest index number among the empty servers. When all servers are busy at a time of an arrival, that arrival must leave the system without being served. The semi-Markov process method is used to describe this model and embedded Markov chain of the process is obtained. Furthermore, the Laplace–Stieltjes transform of the distribution of interoverflow times is derived which is the main objective of the paper. Finally, it is offered a new formulation for the loss probability which provides more efficient and rapid calculation is proposed.

M/M/1/N Queuing System with Retention of Reneged Customers

The concept of customer reneging has been exploited to a great extent in recent past by the queuing modelers. Economically, if we see, the customer reneging leads to loss of potential customers and thereby results into the loss in the total revenue. Taking into consideration this customers’ loss due to reneging, a new queuing model has been developed that deals with customer retention. According to this model, a reneged customer can be convinced in many cases by employing certain convincing mechanism to stay in the queue for completion of his service. Thus, a reneged customer can be retained in the queuing system with some probability (say, q) and it may leave the queue without receiving service with probability p (=1-q). This process is referred to as customer retention. We consider a single server, finite capacity queuing system with customer retention in which the inter-arrival and service times follow negative-exponential distribution. The reneging times are assumed to be exponentially distributed. The steady state solution of the model has been obtained. Some performance measures have been computed. The sensitivity analysis of the model has been carried out. The effect of probability of retention on the average system size has been studied. The numerical results show that the average system size increases steadily as the probability of retention increases. Some particular cases of the model have been derived and discussed.

A Complete and Simple Solution to a Discrete-Time Finite-Capacity BMAP/D/c Queue

We consider a discrete-time multi-server finite-capacity queueing system with correlated batch arrivals and deterministic service times (of single slot), which has a variety of potential applications in slotted digital telecommunication systems and other related areas. For this queueing system, we present, based on Markov chain analysis, not only the steady-state distributions but also the transient distributions of the system length and of the system waiting time in a simple and unified manner. From these distributions, important performance measures of practical interest can be easily obtained. Numerical examples concerning the superposition of certain video traffics are presented at the end.

A note on optimal pricing for finite capacity queueing systems with multiple customer classes

This article investigates optimal static prices for a finite capacity queueing system serving customers from different classes. We first show that the original multi-class formulation in which the price for each class is a decision variable can be reformulated as a single dimensional problem with the total load as the decision variable. Using this alternative formulation, we prove an upper bound for the optimal arrival rates for a fairly large class of queueing systems and provide sufficient conditions that ensure the existence of a unique optimal arrival rate vector. We show that these conditions hold for M/M/1/m and M/G/s/s systems and prove structural results on the relationships between the optimal arrival rates and system capacity. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008