Interdeparture Time Research Articles

Variability propagation in manufacturing systems: the impact of the processing time distribution

ABSTRACT In manufacturing and service systems, the negative effects of variability propagation are usually addressed through approximation models and other tools, such as simulation and optimization algorithms. This paper investigates the conditions in which these approaches are ineffective, and considering only the mean and variance of the processing time distribution is misleading. A scenario analysis deepens the impacts of considering the entire processing time distribution (beyond its mean and variance) on the inter-departure times in balanced and unpaced lines modeled through Discrete Event Simulation. The results correlate the impacts on variability propagation of approximating the processing time distribution with system characteristics such as utilization levels, line sizes, and inter-arrival and processing time variability. Production planning approaches can benefit from these results to reduce variability propagation, particularly in flexible and reconfigurable manufacturing systems, largely adopted in Industry 4.0, that can highly influence processing time distributions by varying product mix and line configuration.

Inter-Departure Time Correlations in PH/G/1 Queues

In non-Markovian tandem queueing networks the output process of one site, which is the input process to the next site, is not renewal. Consequently, the correlation analysis of that output processes is essential when studying such networks. A correlation analysis in the M/G/1 queue has been studied in the literature via derivation of the joint Laplace-Stieltjes transform (LST) of the sum of two consecutive inter-departure times. That LST is obtained by considering all possible cases at departure epochs. However, those epochs are expressed via dependent variables. In this paper, we first extend the analysis to the more general PH/G/1 queue, and investigate various queues, such as E2/G/1 and C2/C2/1. Then, we consider the lag-n correlation, which requires derivation of the joint LST of sum of n + 1 consecutive inter-departure times. Yet, deriving this LST by the common approach becomes impractical for n + 1 ≥ 3, as the number of all possible cases at departure epochs increases significantly. To overcome this obstacle, we derive a corresponding single-parameter LST, which expresses the sum of n + 1 consecutive inter-departure times via the (n + 1)-st departure epoch only. Consequently, the latter LST is expressed via a much fewer number of possible cases, and not less important, as a function of independent variables only, eliminating the need to derive the corresponding joint density. Considering the M/G/1 and the E2/G/1 queues, we demonstrate that the joint LST can be reconstructed directly via the corresponding single-parameter LST when n + 1 = 2. We further conjecture that the multi-parameter joint LST can be reconstructed from the corresponding single-parameter LST in more general queues and for values of n + 1 > 2. The conjecture is validated for various PH/G/1 queues and proved for n + 1 = 3 in the M/G/1 case. The new approach facilitates the calculation of lag-n correlation of the departure process from PH/G/1 queue for n + 1 ≥ 3. Our analysis illuminates the cases when using renewal approximation of the output process provides a proper approximation when studying non-Markovian stochastic networks.

Study of a two types of general heterogeneous service queueing system in a single server with optional repeated service and feedback queue

This paper addresses a model on a single server queue and two service representatives. After a customer is served, he/she has the three options: opting for receive the same service again (re-service), joining as a new customer for another regular service (feedback), or leaving the service system altogether. To ensure the queueing system is Markovian, we introduce an additional variable (supplementary variable) and using this approach, we derive the explicit distribution of queue size at random and departure epochs. Additionally, we determine the distribution of response time, inter-departure time, and busy period. By using the embedded Markov chain technique we have also derived the queue size distribution at departure epoch. We have also presented the cost analysis of the model with some numerical examples. The numerical illustration validates our findings and provides valuable insights into the queuing system.

Marginal and joint distribution of inter-departure times for a PH/PH/c queue

In this paper, we introduce the Laplace–Stieltjes transform (LST) of the inter-departure time distribution in a PH/PH/c queue, and the two-dimensional joint LST of two consecutive inter-departure times to construct their correlation structure. We exploit the properties of phase-type (PH) random variables, as well as the steady-state distribution of the underlying continuous-time Markov chain in a PH/PH/c queue to construct these LSTs. We demonstrate our approach through numerical examples, while validating the results. Later, we analyze the correlation between two consecutive inter-departure times for various PH/PH/c queues. We observe that, if the fundamental elements of the queue have high (low) variability, then the correlation is positive (negative).

Penyesuaian distribusi proses keberangkatan sepeda motor dari lahan parkir saat waktu puncak

The study aims to model the distribution of motorcycle departures from the parking lot at peak times with the Poisson process approach. This process involves a discrete number of departures and continuous interdeparture time. The probability distribution candidate was selected to model the data according to the data nature, the stochastic process, and the empirical observation of the departure process. Parameters are estimated using the Maximum Likelihood Estimation (MLE) method and the bootstrapping procedure to construct confidence intervals for the parameter. The goodness-of-fit test is applied to select the best probability distribution that matches empirical data. Inferences to the distribution parameters suggest that Weibull's distribution is more appropriate for describing the motorcycle's inter-departure time. The number of motorcycle departures fits significantly into a negative binomial distribution. The results of the study concluded that the Poisson process applied was a case of overdispersion, with the motorcycle departure rate decreasing over time.Keywords: Bootstrapping, departure, distribution, goodness-of-fit, PoissonMSC2020: 60E05

Age of error information and throughput for truncated and layer‐coded HARQ‐based satellite‐IoT systems

SummaryThe work focuses on the simultaneous improvement on the throughput and information freshness of the promising satellite‐based Internet of Things (S‐IoT). To this end, we first propose a method based on the adaptive modulation coding (AMC) and truncated layer‐coded hybrid automatic repeat request (L‐HARQ) strategies by especially considering the difference in the channel state information (CSI) between the transmitter and receiver. On account of the large signal delay of the S‐IoT systems, this paper establishes the channel model based on the correlation of the signal‐to‐noise ratio (SNR) under the outdated channel and the ideal channel. Secondly, considering the fact that the improved backtrack decoding is used in the L‐HARQ, the classical definition of age of information (AoI) is insufficient to quantify the status update freshness, we propose the age of error information (AoEI), which is defined as the elapsed time since the generation of the lastly successfully backtrack‐decoded status update. Thirdly, we characterize the statistical descriptions of the maximum wait time of the successfully recovered status updates and the interdeparture time between the two consecutive status update packets that are lastly successfully received. Then, the closed‐form expression of average AoEI is derived under the assumption of independent and identically distributed fading channels. Finally, the Markov decision process (MDP) framework is used to optimize the packet parameters in each round of L‐HARQ process under the general fading scenario. Thereby, the optimized transmission scheme that minimizes AoI with throughput constraint is solved. The simulations investigate the effect of system parameters on the average AoEI.

Correlated Queues with Inter-Arrival Times Depending on Service Times

We consider a type of correlated queue in which the inter-arrival time between two consecutive customers linearly depends on the service time of the first customer. We first derive infinite systems of linear equations for the moments of the waiting time in two special cases, based on which we then develop several methods to calculate the moments of the waiting time by using MacLaurin series approximation, Padé approximation and truncation method. In addition, we show how the moments and covariances of the inter-departure times of the correlated queue can be calculated based on the moments of the waiting time. Finally, extensive numerical examples are provided to validate our methods.

Age of Error Information and Throughput Tradeoff for IoT Status Update Systems With Truncated and Layer-Coded HARQ

A novel status update system is designed by combining adaptive modulation and coding, and truncated and layer-coded hybrid automatic repeat request (T-L-HARQ). With the channel state information (CSI) at transmitter (CSIT), which is obtained from the receiver by using pilot-assisted estimation, the source adaptively selects its transmission rate. Due to the delay, the CSI at receiver (CSIR) is the outdated version of the CSIT, and the decoding error is inevitable. Firstly, since the backtrack decoding used in the T-L-HARQ, the classical age of information is insufficient to quantify the status update freshness, we propose the age of error information (AoEI), which is defined as the elapsed time since the generation of the lastly successfully backtrack-decoded status update. Secondly, we characterize the interdeparture time duration of two consecutively successfully feedforward transmissions and the backtrack decoding depth of a successful T-L-HARQ transmission cycle. As a result, the closed-form expression of average AoEI is derived. The simulations show that a tradeoff design between throughput and AoEI is essential by controlling backtrack decoding depth.

Spatiotemporal synchronous coupling algorithm for urban rail transit timetables design under dynamic passenger demand

This paper focuses on the problem of designing the schedule of urban rail transit systems under dynamic passenger demand considering limited train capacity and congestion. From the point of view of operators, the goal of timetable optimization consists of using as few as possible service trains to transport the arriving passengers to their destinations securely and quickly. On the other hand, passengers want to spend the minimum time travelling to their destinations, including the time they wait on platforms until they board the first and successive trains in case of transferring between lines. Due to the variability of passenger flows and the difficulty in exactly solving full-day timetabling problems in short computation times, it is difficult in practice to use the approaches proposed in the literature. Moreover, when the capacity of trains is considered and oversaturation conditions emerge, the problem becomes even more complex, converting into a challenge the design of schedules from an operational point of view. By considering fixed train dwell times at stations and train running times between stations, we first propose a non-linear programming formulation to model the problem of determining the most convenient interdeparture times of services at the first station of a line, aiming at minimizing the waiting time of passengers. Since the dynamic behaviour of passenger demand and the maximum sectional passenger flow are approximately synchronous, an innovative method, the spatiotemporal synchronization coupling algorithm, based on working with the time-varying maximum sectional passenger flow instead of the demand patterns is proposed to quickly solve in an approximate way the demand-driven timetabling problem. The proposed approach is applied to a real case of Chengdu Metro Line 1 in China. The results show the effectiveness of the algorithm, both in terms of computational effort and in terms of reducing the waiting time currently experienced by passengers.

Estimating Multiclass Service Demand Distributions Using Markovian Arrival Processes

Building performance models for software services in DevOps is costly and error-prone. Accurate service demand distribution estimation is critical to precisely modeling queueing behaviors and performance prediction. However, current estimation methods focus on capturing the mean service demand, disregarding higher-order moments of the distribution that still can largely affect prediction accuracy. To address this limitation, we propose to estimate higher moments of the service demand distribution for a microservice from monitoring traces. We first generate a closed queueing model to abstract software performance and use it to model the departure process of requests completed by the software service as a Markovian arrival process (MAP). This allows formulating the estimation of service demand into an optimization problem, which aims to find the first multiple moments of the service demand distribution that maximize the likelihood of the MAP using generated the measured inter-departure times. We then estimate the service demand distribution for different classes of service with a maximum likelihood algorithm and novel heuristics to mitigate the computational cost of the optimization process for scalability. We apply our method to real traces from a microservice-based application and demonstrate that its estimations lead to greater prediction accuracy than exponential distributions assumed in traditional service demand estimation approaches for software services.

Shall firms withhold exact waiting time information from their customers? A transport example

Apps which allow customers to check waiting times in real time on their smartphones become increasingly popular across many industries. Waiting time information is ambiguous in the sense that it is good news for some customers and bad news for other customers depending on whether they get to know that they have to wait short or long times, respectively. This paper studies the strategic choice of information strategies by considering a transport company that charges a uniform price to passengers. Waiting times are determined by headways which are given by the time difference between vehicle departures. The sum of the price and waiting time costs (determined by the waiting times and time valuations) represent the full price of traveling which vary among passengers depending on arrival times at the station. Passengers respond to variations in the full price by adjusting their demand rates in the sense that passengers may use an alternative mode of transport or refuse to travel if they have to wait or expect to wait for too long for the next departure. Two types of information strategies are considered. The company can advertise the exact schedule or advertise only the average lengths of interdeparture times (headways). The analysis shows that the curvature of the demand-rate function in the waiting time is crucial for the choice of information strategies in both the case of profit-maximizing companies and the case of welfare-maximizing companies where welfare is calculated by the sum of profit and consumer surplus. Numerical examples further indicate the crucial role of demand rate curvatures for headway and price choices.

Supervised-learning-based approximation method for multi-server queueing networks under different service disciplines with correlated interarrival and service times

ABSTRACT Developing efficient performance evaluation methods is important to design and control complex production systems effectively. We present an approximation method (SLQNA) to predict the performance measures of queueing networks composed of multi-server stations operating under different service disciplines with correlated interarrival and service times with merge, split, and batching blocks separated with infinite capacity buffers. SLQNA yields the mean, coefficient of variation, and first-lag autocorrelation of the inter-departure times and the distribution of the time spent in the block, referred as the cycle time at each block. The method generates the training data by simulating different blocks for different parameters and uses Gaussian Process Regression to predict the inter-departure time and the cycle time distribution characteristics of each block in isolation. The predictions obtained for one block are fed into the next block in the network. The cycle time distributions of the blocks are used to approximate the distribution of the total time spent in the network (total cycle time). This approach eliminates the need to generate new data and train new models for each given network. We present SLQNA as a versatile, accurate, and efficient method to evaluate the cycle time distribution and other performance measures in queueing networks.

Supervised learning-based approximation method for single-server open queueing networks with correlated interarrival and service times

Efficient performance evaluation methods are needed to design and control production systems. We propose a method to analyse single-server open queueing network models of manufacturing systems composed of delay, batching, merge and split blocks with correlated interarrival and service times. Our method (SLQNA) is based on using a supervised learning approach to determine the mean, the coefficient of variation, and the first-lag autocorrelation of the inter-departure time process as functions of the mean, coefficient of variation and first-lag autocorrelations of the interarrival and service times for each block, and then using the predicted inter-departure time process as the input to the next block in the network. The training data for the supervised learning algorithm is obtained by simulating the systems for a wide range of parameters. Gaussian Process Regression is used as a supervised learning algorithm. The algorithm is trained once for each block. SLQNA does not require generating additional training data for each unique network. The results are compared with simulation and also with the approximations that are based on Markov Arrival Process modelling, robust queueing, and G/G/1 approximations. Our results show that SLQNA is flexible, computationally efficient, and significantly more accurate and faster compared to the other methods.

Bottleneck detection in high-variety make-to-Order shops with complex routings: an assessment by simulation

This study uses simulation to assess the performance of alternative methods for detecting momentary bottlenecks in high-variety contexts that produce on a to-order basis. The results suggest that using the utilisation level of a station to detect bottlenecks leads to the best performance, but that this method suffers from high nervousness. Using the active period of a station appears to be a better overall choice for practice given its good performance and low nervousness. Meanwhile, methods that focus on the workload at a station are a viable alternative, but they may become dysfunctional in shops with directed routings and a limit on the queue. This negative effect is even stronger if the corrected workload measure is used, as recently suggested in the literature on short term capacity adjustments. Finally, using the inter-departure time detection method leads to the worst performance since: (i) it counterintuitively detects non-bottlenecks instead of bottlenecks; and, (ii) it is based on historical data, leading to a response delay.

Status Update in IoT Networks: Age-of-Information Violation Probability and Optimal Update Rate

The Internet of Things (IoT) has emerged as one of the key features of the next-generation wireless networks, where timely delivery of status update packets is essential for many real-time IoT applications. Age of Information (AoI) is a new metric to measure the freshness of update. The reduction of the violation probability that AoI of status updates exceeds a given age constraint is of great significance for guaranteeing the information freshness in IoT systems. By modeling the IoT networks as M/M/1 and M/D/1 queuing systems, this work focuses on characterizing the violation probability of peak AoI and AoI in IoT systems, where a sensor delivers updates to a monitor under M/M/1 and M/D/1 queues with first-come-first-served policy. From a time-domain perspective, we explore the correlation between interdeparture time and system time, by which the closed-form expressions of peak AoI distribution and the violation probability for any AoI constraint are derived. The obtained results induce accurate characterizations for probability distribution functions of peak AoI and AoI. Consequently, accurate characterizations of average AoI and the variance of AoI are obtained. Then, for peak AoI and AoI, the optimal generation rate of the status update that induces the minimal violation probability is also found. The numerical results show that the optimal update rate can significantly reduce the AoI violation probability for a wide range of AoI constraints. The theoretical findings and predictions are verified by numerical simulation results as well as provide guidance for the design of IoT networks.

A new solution approach for multi-stage semi-open queuing networks: An application in shuttle-based compact storage systems

Multi-stage semi-open queuing networks (SOQNs) are widely used to analyze the performance of multi-stage manufacturing systems and automated warehousing systems. While there are several methods available for solving single-stage SOQNs, solution methods for multi-stage SOQNs are limited. Decomposition of a multi-stage SOQN into single-stage SOQNs and evaluation of an individual single-stage SOQN is a possibility. However, the challenge lies in obtaining the job departure process information from an upstream single-stage SOQN to evaluate the performance of a downstream single-stage SOQN. In this paper, we propose a two-moment approximation approach for estimating the squared coefficient of variation of the job inter-departure time from a single-stage SOQN, which can serve as an input to link multi-stage SOQNs. Using numerical experiments, we test the robustness of the proposed approach for various input parameter settings for both single and multi-class jobs. We find that the proposed approach works quite well, particularly when the coefficient of variation of the job inter-arrival time is less than two. We demonstrate the efficacy of the proposed approach using a case study on a multi-tier shuttle-based compact storage system and benchmark our results with an existing approach. The results indicate that our approach yields more accurate estimates of the performance measures in comparison to the existing approach in the literature.

Reducing the variability of inter-departure times of a single-server queueing system – The effects of skewness

A critical performance measure in serial production lines is the variability of inter-departure times of a single-server queue. Increasing upstream inter-departure time variability generates greater downstream variability, diminishing overall line performance. Theory suggests that the variability of inter-departure times of a single-server queue is reduced by decreasing the variance of inter-arrival and service times. This study investigates the effects of the skewness of inter-arrival and service time distributions on the variability of inter-departure times. Contrary to previous results suggesting that mean waiting times of a GI/G/1 queue can be reduced by increasing inter-arrival time skewness, this experimental study of a GI/G/1 queue with triangular inter-arrival and service times shows that the inter-departure time coefficient of variation is reduced through a combination of negative inter-arrival time skewness and positive service time skewness. These results also suggest that the absolute value of the negative autocorrelation between consecutive departures is reduced by the same combination of negative inter-arrival time skewness and positive service time skewness for low values of server’s utilization, while positive skewness for both inter-arrival and service times reduces this value for high values of server’s utilization. Finally, it was found that queue capacity constraints increase the coefficient of variation of inter-departure times, as has been previously suggested, as well as the skewness and the absolute correlation values of the inter-departure time distribution.

New LST of inter-departure times in PH/G/1 queue, and extensions to ME/G/1 and G/G/1 queues

In this paper, we provide a new approach to model the inter-departure times distribution in a PH/G/1 queue. This approach enables to further model the inter-departure times distribution in more general queues as well. Initially, we propose to express the Laplace–Stieltjes transform (LST) of inter-departure times in PH/G/1 queues by exploiting the probabilistic interpretation of phase-type distributions. Using this interpretation enables to eliminate the necessity of the matrix-geometric method, and thus significantly reduces the computational complexity. Then, we use the LST of inter-departure times distribution in a Cm/G/1 queue to express this LST in a ME/G/1 queue, where ME is a Matrix-Exponential distribution. We validate it in a few ME/G/1 examples. Finally, we propose to approximate the LST of inter-departure times distribution in a G/G/1 queue by employing the above LST of the proper PH/G/1 queue. Without loss of generality, we demonstrate our proposed approximation by using the LST as obtained in a Cm/G/1 queue, while illustrating by a few G/G/1 examples that the accuracy can be as good as one might want.

Degradedness and Secrecy in Memoryless Queues

We study exponential-server timing channels, in which the transmitter encodes a message using a sequence of packet inter-arrival times and the receiver observes the corresponding inter-departure times from an exponential-server queue. We show that if the service rate of one such queue is faster than that of another, then the latter channel is stochastically degraded with respect to the former. This combined with Burke’s theorem implies that the slower queue is more entropy increasing than the first. These facts put together provide a converse for the secrecy capacity of the wiretapped version of the channel, matching an earlier achievability result in the literature.

Serial production line performance under random variation: Dealing with the ‘Law of Variability’

Many Queueing Theory and Production Management studies have investigated specific effects of variability on the performance of serial lines since variability has a significant impact on performance. To date, there has been no single summary source of the most relevant research results concerned with variability, particularly as they relate to the need to better understand the ‘Law of Variability’. This paper fills this gap and provides readers the foundational knowledge needed to develop intuition and insights on the complexities of stochastic simple serial lines, and serves as a guide to better understand and manage the effects of variability and design factors related to improving serial production line performance, i.e. throughput, inter-departure time and flow time, under random variation.