Phase-type Service Time Distribution Research Articles

Investigation of the Fork–Join System with Markovian Arrival Process Arrivals and Phase-Type Service Time Distribution Using Machine Learning Methods

This paper presents a study of fork–join systems. The fork–join system breaks down each customer into numerous tasks and processes them on separate servers. Once all tasks are finished, the customer is considered completed. This design enables the efficient handling of customers. The customers enter the system in a MAP flow. This helps create a more realistic and flexible representation of how customers arrive. It is important for modeling various real-life scenarios. Customers are divided into K≥2 tasks and assigned to different subsystems. The number of tasks matches the number of subsystems. Each subsystem has a server that processes tasks, and a buffer that temporarily stores tasks waiting to be processed. The service time of a task by the k-th server follows a PH (phase-type) distribution with an irreducible representation (βk, Sk), 1≤k≤K. An analytical solution was derived for the case of K=2 when the input MAP flow and service time follow a PH distribution. We have efficient algorithms to calculate the stationary distribution and performance characteristics of the fork–join system for this case. In general cases, this paper suggests using a combination of Monte Carlo and machine learning methods to study the performance of fork–join systems. In this paper, we present the results of our numerical experiments.

Distribution of Age of Information in Status Update Systems With Heterogeneous Information Sources: An Absorbing Markov Chain-Based Approach

In this letter, we obtain the exact distributions of the Age of Information (AoI) and Peak AoI (PAoI) in a non-preemptive multi-source status update system for (i) Generate-At-Will (GAW) servers with probabilistic transmissions (ii) Random Arrival with Single Buffer (RA-SB) servers employing probabilistic buffer management, using absorbing Continuous-Time Markov Chains (CTMC). For both servers, the information sources are allowed to have different relative urgencies, phase-type service time distributions, and transmission error probabilities, for the sake of generality. Numerical examples are presented to validate the proposed analytical model.

Improving the Mean-Field Fluid Model of Processor Sharing Queueing Networks for Dynamic Performance Models in Cloud Computing

Resource management in cloud computing is a difficult problem, as one needs to balance between adequate service to clients and cost minimization in a dynamic environment of interconnected components. To make correct decisions in such an environment, good performance models are necessary. A common modeling methodology is to use networks of queues, but as these are prohibitively expensive to evaluate for many applications, approximation methods for key metrics are frequently employed. One such method-that provides both transient solutions and short, scalable computation times-is the fluid model, which approximates the dynamics of the mean queue lengths using a system of ordinary differential equations. However, finding a fluid model that can adequately approximate an arbitrary queueing network is in general difficult. In the paper, we extend the state of the art with the following three contributions. First, we show that for any mixed multiclass queueing network of processor sharing and delay queues with phase-type service time distributions, such a fluid model can be found via the meanfield approximation. Furthermore, we propose an improved model based on smoothing of the processor share function that improves the performance of certain systems. Finally, using the smoothed mean-field model, we introduce an accurate closed-form approximation of the response time CDF over any subset of classes and queues.

The calibration method of a tandem queueing model with PH service time using NS-3 simulation of a multihop wireless network

End-to-end delay is one of the key characteristics of communication network performance. This characteristic determines the possibility of using the network for various delay-critical applications like voice or video transmission. One of the widely used approaches to estimating delays is the use of the queuing theory. According to this approach, a telecommunication network is modeled using a multiphase queuing system. Communication channels are modeled using service devices, and the incoming traffic is modeled with random distributions of the inter-arrival intervals between packets. The accuracy of this network model directly depends on how well the service time distributions are chosen. These distributions must consider the specifics of complex telecommunication protocols, size distributions of the transmitted packets, and, in case of wireless channels, the rate of collisions and retransmissions. The paper presents a study of the accuracy of estimates of end-to-end delays in a multi-hop wireless network using a queuing network with a phase-type (PH) service time distributions. To calibrate the model, PH distributions are found using the moments-matching method based on sample data on the duration of packet transmission in IEEE 802.11 channels. This sample data was obtained using a simulation model written in NS-3, taking into account the features of the IEEE 802.11 protocol and the presence of collisions in the network. To evaluate the accuracy, end-to-end delays are calculated using the queuing network and the wireless network simulation model. It is shown that it is possible to obtain reasonably accurate estimates for small networks, but with an increase in the size of the network, the accuracy decreases. In conclusion, recommendations are given to improve the accuracy of modeling.

Improving the mean-field fluid model of processor sharing queueing networks for dynamic performance models in cloud computing

Resource management in cloud computing is a difficult problem, as one is often tasked with balancing between adequate service to clients and cost minimization in dynamic environments of many interconnected components. To make correct decisions in these environments, good performance models are necessary. A common modeling methodology is to use networks of queues, but as these are prohibitively expensive to evaluate for many real-time applications, different approximation methods for important metrics are frequently employed. One such method—that provides both transient solutions and short, scalable computation times—is the fluid model, which approximates the dynamics of the mean queue lengths using a system of ordinary differential equations. However, finding a fluid model that can adequately approximate an arbitrary queueing network is in general difficult. In this paper, we extend the state of the art with the following three contributions. First, we show that for any mixed multiclass queueing network of processor sharing and delay queues with phase-type service time distributions, such a fluid model can be found via the mean-field approximation. Furthermore, we propose an improved model based on smoothing of the processor share function that improves the performance of certain systems. Finally, using the smoothed mean-field model, we introduce an accurate closed-form approximation of the response time CDF over any subset of classes and queues. The contributions are further evaluated in a large simulation experiment, which shows that they can be used to accurately predict performance metrics under some system perturbations common in cloud computing.

Discrete-Time Queueing Model of Age of Information With Multiple Information Sources

Information freshness in IoT-based status update systems has recently been studied through the Age of Information (AoI) and Peak AoI (PAoI) performance metrics. In this article, we study a discrete-time server arising in multisource IoT systems, which accepts incoming information packets from multiple information sources so as to be forwarded to a remote monitor for status update purposes. Under the assumption of Bernoulli information packet arrivals and a common general discrete phase-type service time distribution across all the sources, we numerically obtain the exact per-source distributions of AoI and PAoI in matrix-geometric form for three different queueing disciplines: 1) nonpreemptive bufferless; 2) preemptive bufferless; and 3) nonpreemptive single buffer with replacement. The proposed numerical algorithm employs the theory of discrete-time Markov chains of quasi-birth-death type and is matrix analytical. Numerical examples are provided to validate the accuracy and effectiveness of the proposed queueing model. We also present a numerical example on the optimum choice of the Bernoulli parameters in a practical IoT system with two sources with diverse AoI requirements.

Stationary queue and server content distribution of a batch-size-dependent service queue with batch Markovian arrival process:

Queueing systems with batch Markovian arrival process (BMAP) have paramount applications in the domain of wireless communication. The BMAP has been used to model the superposition of video sources and to approximate the super-position of data, voice and video traffic. This article analyzes an infinite-buffer generally distributed batch-service queue with BMAP, general bulk service (a, b) rule and batch-size-dependent service time. In this proposed analysis, we mainly focus on deriving the bivariate vector generating function of queue and server content distribution together at departure epoch using supplementary variable technique. The mathematical procedure for the complete extraction of distribution at departure epoch has been discussed and using those extracted probabilities, we achieve the queue and server content distribution at arbitrary epoch. Finally, numerical illustrations have been carried out in order to make a deep insight to the readers which contains deterministic as well as phase-type service time distributions.

Fluid approximation of closed queueing networks with discriminatory processor sharing

As a multi-class variant of the classical egalitarian processor-sharing (EPS) discipline, discriminatory processor sharing (DPS) provides a suitable paradigm to model systems where share exists to control the service access of heterogeneous jobs. Although DPS is a more fine-grained scheduling discipline than EPS, the behavior of closed queueing networks (QNs) with DPS remains poorly understood. We propose in this paper a fluid approach to transient and steady-state analysis of closed QNs comprising delay and DPS stations. Our reference model features an arbitrary routing topology, phase-type service time distributions and class switching. The proposed approach has been validated against simulation for both transient and steady-state analysis. Moreover, we introduce a refined method for approximating response time distributions at either station or system level through transient analysis, showing its improvement upon prior art in handling non-exponentially distributed service times.

First-come-first-served queues with multiple servers and customer classes

We present a simple approach to the solution of a multi-server FCFS queueing system with several classes of customers and phase-type service time distributions. The proposed solution relies on solving a single two-class model in which we distinguish one of the classes and we aggregate the remaining customer classes. We use a reduced state approximation to solve this two-class model. We propose two types of aggregation: exact, in which we merge the phase-type service time distributions exactly, and approximate, in which we simplify the phase-type distribution for the aggregated class by matching only its first two moments. The proposed approach uses simple mathematics and is highly scalable in terms of the number of servers, the number of classes, as well as the number of phases per class. Our approach applies both to queues with finite and infinite buffer space.

Analysis of an infinite-buffer batch-size-dependent service queue with Markovian arrival process

This paper considers an infinite-buffer batch-service queue with Markovian arrival process, generally distributed and batch-size-dependent service time. We obtain a bivariate vector generating function of queue length and server content distribution at departure epoch of a batch. The complete joint distribution of queue length, server content and phase of the arrival process at departure epoch is extracted in terms of roots of the associated characteristic equation. By employing these probability vectors we also perceive the joint distribution at arbitrary and pre-arrival epochs. Our analytic procedure and results are demonstrated using some numerical examples for phase-type as well as deterministic service time distributions with high and low values of model parameters. The occurrence of multiple roots are also investigated in case of phase-type service time distribution. Finally, we also investigate the influence of correlation of the arrival process on the behavior of the system performance.

On the output dynamics of production systems subject to blocking

ABSTRACTAnalyzing the output dynamics of a production system gives valuable information for operation and performance evaluation of a production system. In this article, we present an analytical method to determine the autocorrelation of the inter-departure times in queueing networks subject to blocking that can be represented by a Continuous-Time Markov Chain. We particularly focus on production systems that are modeled as open or closed queueing networks, and where stations have phase-type service time distributions. We use the analytical results for the mean and the variance of the time to produce a given number of products in queueing networks to determine the correlation of inter-departure times with different lags. We present a computationally efficient recursive method to determine the correlation of the inter-departure times in open and closed queueing networks. The method also yields closed-form expressions for the correlations of a two-station production line with exponential servers and a finite buffer. We show how the correlations develop with increasing lags subject to different processing time distributions, buffer capacities, and number of stations, in both open and closed queueing networks. As a result, we propose the analytical method given in this study as a tool to study the effects of design and control parameters on the output dynamics of production systems.

On the Exact Inter-departure, Inter-start, and Cycle Time Distribution of Closed Queueing Networks Subject to Blocking

This paper presents a method to determine the exact inter-departure, inter-start and cycle time distribution of closed queueing networks that can be modeled as Continuous-Time Markov Chains with finite state space. The method is based on extending the state space to determine the transitions that lead to a departure or to an arrival of a part on a station. Once these transitions are identified and represented in an indicator matrix, a first passage time analysis is utilized to determine the exact distributions of the inter-departure, inter-start, and cycle time. In order to demonstrate the methodology, we consider closed-loop production lines with phase-type service time distributions and finite buffers. We discuss the methodology to automatically generate the state space and to obtain the transition rate matrices for the considered distributions. We use the proposed method to analyze the effects of the system parameters on the inter-departure, inter-start time, and cycle time distributions numerically for various cases. The proposed methodology allows the exact analysis of the inter-departure, inter-start, and cycle time distributions of a wide range of production systems with phase-type servers that can be modeled as Continuous-Time Markov Chains in a unified way.

Analysis of an MMAP/PH1, PH2/N/∞ queueing system operating in a random environment

Abstract A multi-server queueing system with two types of customers and an infinite buffer operating in a random environment as a model of a contact center is investigated. The arrival flow of customers is described by a marked Markovian arrival process. Type 1 customers have a non-preemptive priority over type 2 customers and can leave the buffer due to a lack of service. The service times of different type customers have a phase-type distribution with different parameters. To facilitate the investigation of the system we use a generalized phase-type service time distribution. The criterion of ergodicity for a multi-dimensional Markov chain describing the behavior of the system and the algorithm for computation of its steady-state distribution are outlined. Some key performance measures are calculated. The Laplace-Stieltjes transforms of the sojourn and waiting time distributions of priority and non-priority customers are derived. A numerical example illustrating the importance of taking into account the correlation in the arrival process is presented

Multi-server queueing system with a generalized phase-type service time distribution as a model of call center with a call-back option

A multi-server queueing system with a Markovian arrival process and finite and infinite buffers to model a call center with a call-back option is investigated. If all servers are busy during the customer arrival epoch, the customer may leave the system forever or move to the buffer (such a customer is referred to as a real customer), or, alternatively, request for call-back (such a customer is referred to as a virtual customer). During a waiting period, a real customer can be impatient and may leave the system without service or request for call-back (becomes a virtual customer). The service time of a customer and the dial time to a virtual customer for a server have a phase-type distribution. To simplify the investigation of the system we introduce the notion of a generalized phase-type service time distribution. We determine the stationary distribution of the system states and derive the Laplace–Stieltjes transforms of the sojourn and waiting time distributions for real and virtual customers. Some key performance measures are calculated and numerical results are presented.

Validity of heavy-traffic steady-state approximations in many-server queues with abandonment

We consider $$GI/Ph/n+M$$ G I / P h / n + M parallel-server systems with a renewal arrival process, a phase-type service time distribution, $$n$$ n homogenous servers, and an exponential patience time distribution with positive rate. We show that in the Halfin---Whitt regime, the sequence of stationary distributions corresponding to the normalized state processes is tight. As a consequence, we establish an interchange of heavy-traffic and steady-state limits for $$GI/Ph/n+M$$ G I / P h / n + M queues.

Tandem queueing system with infinite and finite intermediate buffers and generalized phase-type service time distribution

A tandem queueing system with infinite and finite intermediate buffers, heterogeneous customers and generalized phase-type service time distribution at the second stage is investigated. The first stage of the tandem has a finite number of servers without buffer. The second stage consists of an infinite and a finite buffers and a finite number of servers. The arrival flow of customers is described by a Marked Markovian arrival process. Type 1 customers arrive to the first stage while type 2 customers arrive to the second stage directly. The service time at the first stage has an exponential distribution. The service times of type 1 and type 2 customers at the second stage have a phase-type distribution with different parameters. During a waiting period in the intermediate buffer, type 1 customers can be impatient and leave the system. The ergodicity condition and the steady-state distribution of the system states are analyzed. Some key performance measures are calculated. The Laplace–Stieltjes transform of the sojourn time distribution of type 2 customers is derived. Numerical examples are presented.

Retrial queuing system with Markovian arrival flow and phase-type service time distribution

We consider a multi-server queuing system with retrial customers to model a call center. The flow of customers is described by a Markovian arrival process (MAP). The servers are identical and independent of each other. A customer’s service time has a phase-type distribution (PH). If all servers are busy during the customer arrival epoch, the customer moves to the buffer with a probability that depends on the number of customers in the system, leaves the system forever, or goes into an orbit of infinite size. A customer in the orbit tries his (her) luck in an exponentially distributed arbitrary time. During a waiting period in the buffer, customers can be impatient and may leave the system forever or go into orbit. A special method for reducing the dimension of the system state space is used. The ergodicity condition is derived in an analytically tractable form. The stationary distribution of the system states and the main performance measures are calculated. The problem of optimal design is solved numerically. The numerical results show the importance of considering the MAP arrival process and PH service process in the performance evaluation and capacity planning of call centers.

Computation of the steady state distribution for multi-server retrial queues with phase type service process

We consider a multi-server retrial queueing system with the Batch Markovian Arrival Process and phase type service time distribution. Such a general queueing system suits for modeling and decision making in many real life objects including modern wireless communication networks. Behavior of such a system is described by the level dependent multi-dimensional Markov chain. Blocks of the generator of this chain, which is the block structured matrix of infinite size, can have large size if the number of servers is large and distribution of service time is not exponential. Due to this fact, the existing in literature algorithms allow to compute key performance measures of such a system only for a small number of servers. Here we describe the algorithm that allows to compute the stationary distribution of the system for larger number of servers and numerically illustrate its advantage. Importance of taking into account correlation in the arrival process is numerically demonstrated.

Many-server diffusion limits for G/Ph/n+GI queues

This paper studies many-server limits for multi-server queues that have a phase-type service time distribution and allow for customer abandonment. The first set of limit theorems is for critically loaded $G/Ph/n+GI$ queues, where the patience times are independent and identically distributed following a general distribution. The next limit theorem is for overloaded $G/ Ph/n+M$ queues, where the patience time distribution is restricted to be exponential. We prove that a pair of diffusion-scaled total-customer-count and server-allocation processes, properly centered, converges in distribution to a continuous Markov process as the number of servers $n$ goes to infinity. In the overloaded case, the limit is a multi-dimensional diffusion process, and in the critically loaded case, the limit is a simple transformation of a diffusion process. When the queues are critically loaded, our diffusion limit generalizes the result by Puhalskii and Reiman (2000) for $GI/Ph/n$ queues without customer abandonment. When the queues are overloaded, the diffusion limit provides a refinement to a fluid limit and it generalizes a result by Whitt (2004) for $M/M/n/+M$ queues with an exponential service time distribution. The proof techniques employed in this paper are innovative. First, a perturbed system is shown to be equivalent to the original system. Next, two maps are employed in both fluid and diffusion scalings. These maps allow one to prove the limit theorems by applying the standard continuous-mapping theorem and the standard random-time-change theorem.

Semi-linear Stochastic Difference Equations

We consider in this paper a class of vector valued processes that have the form Y n + 1 = A n ( Y n ) + B n . B n is assumed to be stationary ergodic and A n is assumed to have a divisibility property. This class includes linear stochastic difference equations as well as multi-type branching processes (with a discrete or with a continuous state space). We derive explicit expressions for the probability distribution as well as for the two first moments of state vectors at the stationary regime. We then apply this approach to derive two formalisms to describe the infinite server queue. The first is based on a branching process approach adapted to phase type service time distributions. The second is based on a linear stochastic difference equation and is adapted to independent and generally distributed service times with bounded support. In both cases we allow for generally distributed arrival process (not necessarily i.i.d. nor Markovian).