Mean Queue Size Research Articles

A Bayesian approach to determine sample size in Erlang single server queues

An essential aspect of the practical application of Erlang single-server queues is the statistical inference of their parameters, notably traffic intensity. This metric is crucial as it serves as a fundamental performance indicator, allowing the derivation of other significant measures, including the mean queue size and the expected number of customers in the system. Additionally, it provides the proportion of time the queue system is occupied. This article explores algorithms for calculating sample sizes for these estimations, based on the number of customers who arrived during service periods, a highly intuitive approach, allowing data collection without concerns about correlations among data points. For this purpose, we have considered two forms of informative priors: Gauss hypergeometric and beta priors. Additionally, a non conjugate and objective prior, known as Jeffreys prior, has been taken into account. We present tables with sample sizes for specific configurations and abacuses for more general configurations obtainable through approximate interpolations.

A 3-queue polling system with join the shortest-serve the longest policy

In 1987, J.W. Cohen analyzed the so-called Serve the Longest Queue (SLQ) queueing system, where a single server attends two non-symmetric M/G/1-type queues, exercising a non-preemptive priority switching policy. Cohen further analyzed in 1998 a non-symmetric 2-queue Markovian system, where newly arriving customers follow the Join the Shortest Queue (JSQ) discipline. The current paper generalizes and extends Cohen’s works by studying a combined JSQ–SLQ model, and by broadening the scope of analysis to a non-symmetric 3-queue system, where arriving customers follow the JSQ strategy and a single server exercises the preemptive priority SLQ discipline. The system states’ multi-dimensional probability distribution function is derived while applying a non-conventional representation of the underlying process’s state-space. The analysis combines both Probability Generating Functions and Matrix Geometric methodologies. It is shown that the joint JSQ–SLQ operating policy achieves extremely well the goal of balancing between queue sizes. This is emphasized when calculating the Gini Index associated with the differences between mean queue sizes: the value of the coefficient is close to zero. Extensive numerical results are presented.

Monotonicity of the mean queue size in DPS systems

Monotonicity of the mean queue size in DPS systems

Performance evaluation in BRT systems: An analysis to predict the BRT systems planning

Mobility issues may create various inconveniences in urban areas due to traffic light delays, traffic jams and acidents. Alternatives have been created to improve public transportation. One of them is the Bus rapid transit system (BRT), delivering convincing results from a cost-effective perspective; i.e., lower deployment costs and a greater number of passengers. This paper proposes a Stochastic Petri Net (SPN) model for performance evaluation of the BRT system, focusing on the mean system size, mean queue size, mean queue time and the probability that the user will miss the bus (discard probability). Scenarios based on the BRT system were created, improving a set of solutions being with a variation in headways and in the number of vehicles on the route. The results show that, from a management perspective, it is up to the decision maker to define which is the most important metric at a given moment and, thus, to define the start intervals, as well as the number of vehicles to have on the route. Under the perspective in study, the best scenario is presented with headways of 300s and 5 vehicles in the route, operating as the one that leads to lower waiting times for passengers.

Modeling and Evaluation of Multi-Hop Wireless Networks Using SRNs

As multi-hop wireless networks are attracting more attention, the need to evaluate their performance becomes essential. In order to evaluate the performance metrics of multi-hop wireless networks, including sending and receiving rates of a node as well as the collision probability, a model based on Stochastic Reward Nets (SRNs) is proposed. The proposed SRN models a typical node in such networks, considered as a general template to be applied to any wireless node. The SRN model of a single node is designed to take transmission effects of all neighboring nodes into account, while ignoring the ones whose transmission has no effect on the node under-study. Applying the proposed SRN to each node in a multi-hop wireless network, we can model any arbitrary topology and evaluate the mean queue size, collision probability, sending and receiving rates of the nodes. Common issues related to Medium Access Control (MAC) and physical layer of wireless networks, e.g. hidden and exposed node problems, collision, transmission, interference and carrier sense ranges, are considered in the proposed SRN model. The results obtained by the proposed model, in two different scenarios, are validated with the results obtained from the discrete-event simulation conducted in ns-2 simulator to show the applicability and accuracy of the proposed SRN model.

Customer mobility and congestion in supermarkets.

The analysis and characterization of human mobility using population-level mobility models is important for numerous applications, ranging from the estimation of commuter flows in cities to modeling trade flows between countries. However, almost all of these applications have focused on large spatial scales, which typically range between intracity scales and intercountry scales. In this paper, we investigate population-level human mobility models on a much smaller spatial scale by using them to estimate customer mobility flow between supermarket zones. We use anonymized, ordered customer-basket data to infer empirical mobility flow in supermarkets, and we apply variants of the gravity and intervening-opportunities models to fit this mobility flow and estimate the flow on unseen data. We find that a doubly-constrained gravity model and an extended radiation model (which is a type of intervening-opportunities model) can successfully estimate 65%-70% of the flow inside supermarkets. Using a gravity model as a case study, we then investigate how to reduce congestion in supermarkets using mobility models. We model each supermarket zone as a queue, and we use a gravity model to identify store layouts with low congestion, which we measure either by the maximum number of visits to a zone or by the total mean queue size. We then use a simulated-annealing algorithm to find store layouts with lower congestion than a supermarket's original layout. In these optimized store layouts, we find that popular zones are often in the perimeter of a store. Our research gives insight both into how customers move in supermarkets and into how retailers can arrange stores to reduce congestion. It also provides a case study of human mobility on small spatial scales.

Interference queueing networks on grids

Consider a countably infinite collection of interacting queues, with a queue located at each point of the $d$-dimensional integer grid, having independent Poisson arrivals, but dependent service rates. The service discipline is of the processor sharing type, with the service rate in each queue slowed down, when the neighboring queues have a larger workload. The interactions are translation invariant in space and is neither of the Jackson Networks type, nor of the mean-field type. Coupling and percolation techniques are first used to show that this dynamics has well-defined trajectories. Coupling from the past techniques are then proposed to build its minimal stationary regime. The rate conservation principle of Palm calculus is then used to identify the stability condition of this system, where the notion of stability is appropriately defined for an infinite dimensional process. We show that the identified condition is also necessary in certain special cases and conjecture it to be true in all cases. Remarkably, the rate conservation principle also provides a closed-form expression for the mean queue size. When the stability condition holds, this minimal solution is the unique translation invariant stationary regime. In addition, there exists a range of small initial conditions for which the dynamics is attracted to the minimal regime. Nevertheless, there exists another range of larger though finite initial conditions for which the dynamics diverges, even though stability criterion holds.

A Hybrid Framework for TCP Incast Congestion Control in Data Center Networks

Cloud data centers with large bandwidth and low latency networks experiences many-to-one traffic pattern called TCP-incast. It occurs in the partition-aggregate architecture and causes emergence congestion at the network wharfconnected to the parent server, overcoming the port emergence buffer. The ending packet loss requires nodes to encounter loss, retransmit data and slowly rise up throughput per definitive TCP behavior. This paper proposes Receiver-oriented Congestion Control with Edge computing approach (RCCE) for enhancing the speed, nature and firmness of traffic performance. Receiver-oriented Congestion Control (RCC) combines both closed and open loop congestion controls at receiverwhereas edge computing involves localization of traffic management in the middle-tier aggregator for reducing Flow Completion Times (FCT) and latency for the entire application processing deployments. In addition, the centralized controller at the edge balances the load during incast by using spanning trees in a well-made manner by implementing multi-stage Clos networks. The entire prototype is implemented in ns3 and simulation results demonstrates that RCCE has an average decrease of 60.2 % in the 99th percentage latency and 50.4 % of mean queue size in the heavy traffic over TCP.

Mathematical modelling of a tollbooth system with two parallel skill-based servers and two vehicle types

This paper considers a tollbooth system with two parallel heterogeneous servers and two vehicle types (say, cars and trucks), where one server collects tolls from vehicles of both types and the other only serves one type of them. With such characteristic, the system is referred to as “skill-based servers” for brevity in this paper. Meanwhile, vehicles are accommodated in a single common lane and get served on the “global First-Come-First-Served” basis. In fact, such a system and its variants are commonly encountered and their performance measures are of great significance to managers. We first develop a Quasi-Birth-Death process to explicitly model this tollbooth system. Then by applying spectral expansion technique, we derive the stationary probabilities for the computation of the system’s performance measures, such as mean queue size and the idle probability of each server. The Laplace–Stieltjes transforms of an arbitrary vehicle’s sojourn time is derived as well. Finally, numerical results are presented to show the impact of parameters’ selection upon performance measures, and they have intriguing managerial implications. The results also reveal that the tollbooth system with skill-based servers is much more efficient compared to the system with dedicated servers, which has been studied by Melange et al. (Comput Oper Res 71:23–33, 2016), especially in rush hours.

Alternating server with non-zero switch-over times and opposite-queue threshold-based switching policy

A single server alternates between two Markovian queues with non-zero switch-over times. The server’s switching instants are determined by the number of customers accumulated at the unattended queue. Specifically, when queue i(i=1,2) is attended and the number of customers in queue j(j=1,2;j≠i) reaches a threshold, the server starts an exponentially distributed switch-over time to queue j, unless the number of customers in queue i is equal to or above queue i’s threshold. However, if during a switch-over period from queue i to queue j the former reaches its threshold, the switch-over is aborted, and the server immediately returns to queue i and continues to serve the customers there. We analyze the system mainly via Matrix Geometric (MG) methods while deriving explicitly the rate matrix R, and thus eliminating the need for successive substitutions. We further reveal connections between the entries of R and the roots of polynomials related to the Probability Generating Functions (PGFs) of the system states. Expressions for the system’s performance measures are obtained (e.g. mean queue size and mean sojourn time in queue 1, PGF and mean of the queue size in queue 2, as well as the Laplace Stieltjes transform and mean of the sojourn time in queue 2). Numerical results are presented and the effects of the various parameters, as well as the switch-over times, on the performance measures are examined. Seemingly counter-intuitive phenomena are discussed. Finally, various extreme cases are investigated.

Analysis of queues in a random environment with impatient customers

We study M/M/c queues (c = 1, 1 < c < ∞ and c = ∞) in a Markovian environment with impatient customers. The arrivals and service rates are modulated by the underlying continuous-time Markov chain. When the external environment operates in phase 2, customers become impatient. We focus our attention on the explicit expressions of the performance measures. For each case of c, the corresponding probability generating function and mean queue size are obtained. Several special cases are studied and numerical experiments are presented.

Throughput optimization of TCP incast congestion control in large-scale datacenter networks

The many-to-one traffic pattern in datacenter networks leads to Transmission Control Protocol (TCP) incast congestion and puts unprecedented pressure to cloud service providers. The abnormal TCP behaviors in incast increase system response time and unavoidably reduce the applicability of cloud-based system deployments. This paper proposes Receiver-oriented Congestion Control (RCC) to address heavy incast in large-scale datacenter networks. RCC is motivated by oscillatory queue size of switch when handling heavy incast and substantial potential of receiver in congestion control when using TCP. RCC makes effective use of centralized scheduler and Explicit Congestion Notification (ECN) at receiver. The RCC prototype is realized in network simulator 3 (ns3) which implements TCP exactly. This paper details the RCC design and evaluates its performance in diverse and heavy workloads. The evaluation results indicate that RCC has an average decreases of 47.5% in the mean queue size and 51.2% in the 99th-percentile latency in the heavy incast over TCP.

Условное среднее значение очередей в системах массового обслуживания с пакетными потоками заявок

This work considers queuing systems with packet flows of demands featured for modern multiservice telecommunication networks. It presents Pollaczek-Khintchine formula generalization for systems with common type flows. We considered mean queue size dependences under low duty ratio. This work introduces a new term of conditional average value of queue, which is mean value under processor no-downtime condition, and presents algorithm for approximation of queue size dependences in queuing systems with packet flows of demands. Time delay in queues was considered. We defined duty limits, which provide desired queue size and time delays.

Adaptive mean queue size and its rate of change: queue management with random dropping

The random early detection active queue management (AQM) scheme uses the average queue size to calculate the dropping probability in terms of minimum and maximum thresholds. The effect of heavy load enhances the frequency of crossing the maximum threshold value resulting in frequent dropping of the packets. An adaptive queue management with random dropping algorithm is proposed which incorporates information not just about the average queue size but also the rate of change of the same. Introducing an adaptively changing threshold level that falls in between lower and upper thresholds, our algorithm demonstrates that these additional features significantly improve the system performance in terms of throughput, average queue size, utilization and queuing delay in relation to the existing AQM algorithms.

FINITE TWO LAYERED QUEUEING SYSTEMS

We study layered queueing systems comprised two interlacing finite M/M/• type queues, where users of each layer are the servers of the other layer. Examples can be found in file sharing programs, SETI@home project, etc. Let Li denote the number of users in layer i, i=1, 2. We consider the following operating modes: (i) All users present in layer i join forces together to form a single server for the users in layer j (j≠i), with overall service rate μjLi (that changes dynamically as a function of the state of layer i). (ii) Each of the users present in layer i individually acts as a server for the users in layer j, with service rate μj.These operating modes lead to three different models which we analyze by formulating them as finite level-dependent quasi birth-and-death processes. We derive a procedure based on Matrix Analytic methods to derive the steady state probabilities of the two dimensional system state. Numerical examples, including mean queue sizes, mean waiting times, covariances, and loss probabilities, are presented. The models are compared and their differences are discussed.

Time-dependent and steady state solution of an M[x]/(G)/1 queueing system with server's long and short vacations

We study the time-dependent and the steady state behaviour of an M[x]/(G)/1 queue with server’s long and short vacations. Customers arrive at the system in a Poisson stream in batches of variable size. The service time of a customer served by the server is assumed to have a general distribution. At each service completion epoch, the server may opt to take a long vacation with probability p1 and a short vacation with probability p2 or else with probability p3 (p1+p2+p3=1) he may opt to continue to be available in the system. The long and short vacation periods of the server are assumed to have general distributions with different mean vacation times. We obtain time-dependent probability generating functions for the queue size. The corresponding steady state results including the stability condition, steady state probabilities of different states of the system, the mean queue size and mean waiting time in the queue as well as in the system have been derived in explicit and closed forms. Several particular cases of interest including some earlier known results have been derived. Many numerical examples confirming validity of the theoretical results have been peformed.

Modeling and Analysis of the Rotor Blade Refurbishment Process at the Corpus Christi Army Depot

Much of the Army’s equipment is coming to the end of its planned life cycle. At the same time, the Department of Defense and the Army are facing severe budget reductions for the foreseeable future. As a result, the planned modernization and acquisition of new equipment will be delayed. The Army is now forced to keep and maintain current equipment as opposed to retiring old systems and buying new ones. With the increased investment in the current systems, the organizations and depots that maintain and refurbish the Army’s equipment are becoming increasingly valuable assets. Corpus Christi Army Depot (CCAD) is the Army’s only facility for repair and overhaul of rotary wing aircraft. CCAD receives approximately 10 rotor blades per day for the Black Hawk helicopter. Each blade is routed through a detailed inspection and rework process consisting of approximately 67 sequential operations which take approximately 45 days per blade. Recently CCAD has expanded and reorganized the rotor blade refurbishment facility which provides an opportunity to re-examine processes, adjust positioning of work stations, and improve efficiency. In this research we develop a discrete-event simulation model of the CCAD rotor blade refurbishment process in order to identify inefficiencies and examine “what if” scenarios to improve key performance metrics. The key performance metrics used to analyze model input include throughput, work in progress, mean queue time, mean queue size, and workstation utilization. The baseline model revealed that there were two crucial bottlenecks that severely limited the throughput and overall performance of the refurbishment process. Adjusting the capacities of these workstations was very effective in reducing the number of blades in WIP and reducing the impact of the queues in front of these stations, but failed to increase the throughput to the desired amount. Additionally, we found that the loss of one whirl tower’s production would not be a significant factor for CCAD’s performance in terms of throughput since operating with only one whirl tower did not significantly impact metrics of interest for the process.

AN M/M/1 QUEUE IN RANDOM ENVIRONMENT WITH DISASTERS

We study a M/M/1 queue in a multi-phase random environment, where the system occasionally suffers a disastrous failure, causing all present jobs to be lost. The system then moves to a repair phase. As soon as the system is repaired, it moves to phase i with probability qi ≥ 0. We use two methods of analysis to study the probabilistic behavior of the system in steady state: (i) via probability generating functions, and (ii) via matrix geometric approach. Due to the special structure of the Markov process describing the disaster model, both methods lead to explicit results, which are related to each other. We derive various performance measures such as mean queue sizes, mean waiting times, and fraction of lost customers. Two special cases are further discussed.

Queues with slow servers and impatient customers

Abstract We study M / M / c queues ( c = 1 , 1 c ∞ and c = ∞ ) in a 2-phase (fast and slow) Markovian random environment, with impatient customers. The system resides in the fast phase (phase 1) an exponentially distributed random time with parameter η and the arrival and service rates are λ and μ, respectively. The corresponding parameters for the slow phase (phase 0) are γ, λ 0 , and μ 0 ( ⩽ μ ) . When in the slow phase, customers become impatient. That is, each customer, upon arrival, activates an individual timer, exponentially distributed with parameter ξ. If the system does not change its environment from 0 to 1 before the customer’s timer expires, the customer abandons the queue never to return. We concentrate on deriving analytic solutions to the queue-length distributions. We derive, for each case of c, the corresponding probability generating function, and calculate the mean queue size. Several extreme cases are investigated and numerical results are presented.

On the multi-phase M/G/1 queueing system with random feedback

In this paper we consider an M/G/1 queue with k phases of heterogeneous services and random feedback, where the arrival is Poisson and service times has general distribution. After the completion of the i-th phase, with probability θ i the (i + 1)-th phase starts, with probability p i the customer feedback to the tail of the queue and with probability 1 − θ i − p i = q i departs the system if service be successful, for i = 1, 2 , . . . , k. Finally in kth phase with probability p k feedback to the tail of the queue and with probability 1 − p k departs the system. We derive the steady-state equations, and PGF’s of the system is obtained. By using them the mean queue size at departure epoch is obtained.