Matrix-geometric Techniques Research Articles

Understanding the Performance of Flexible Functional Split in 5G vRAN Controllers: A Markov Chain-Based Model

We study Flexible Functional Split functionality of 5G vRAN controllers in 5G networks. We propose an innovative model, based on a Markov Chain, which can be used to characterize their performance. We consider both infinite and finite-buffer controllers. In the former, frames would not be lost (provided the system works in a stable regime), and we thus focus on the time frames stay at the controller. For the finite-buffer controller, there might be losses, and we analyze the trade-off between time at the controller (which might hinder the stringent delay requirements of 5G services), and loss probability. Matrix-geometric techniques are used to resolve the corresponding Quasi-Birth-Death process. The validity of the proposed model is assessed by means of an extensive experiment campaign carried out over an ad-hoc event-driven simulator, which is also used to broaden the analysis, considering different service rate distributions, as well as the variability of the studied performance indicators. The results show that the proposed model can be effectively exploited to tackle the dimensioning of these systems, as it sheds light on how their configuration impacts the expected delay and loss rate.

Analysis of Retrial Queueing-Inventory System with Stock Dependent Demand Rate: (s, S) Versus (s, Q) Ordering Policies

Looking into the common real-life observation, particularly, the continuous time Markovian queueing inventory models with the demand rate depends on the on-hand inventory level. It may influence to gain more sale and profit of a particular item. According to this fact, we consider the demand rate of primary as well as retrial customers depending on the on-hand inventory level. An arriving primary customer finds the positive inventory, immediately they purchase an item with a rate depending on the present stock level and at the time of zero inventory, they will enter into the orbit of infinite capacity. The same assumption also holds for retrial customers if the inventory level(IL) is positive. Based on the ordering policies, we present two models: (i) (s, S) policy (ii) (s, Q) Policy. By applying Neut’s Matrix Geometric techniques, the system characteristics are computed under the steady state conditions and the numerical illustrations show the flexibility of the ordering polices in the proposed model.

QoE Enhanced Mobile Data Offloading With Balking

In this letter, we analyze the delayed mobile data offloading scheme by modeling it as an alternating regenerative process. We use a novel way for deriving transmission delay and offloading efficiency, by analyzing the stochastic process characterizing the virtual waiting time. Furthermore, we propose a delayed offloading scheme with balking to enhance quality of experience. The balking scheme is characterized based on 1) current WLAN status; 2) number of packets queued for transmission; and 3) packet deadline. The proposed scheme achieves reduction in mean transmission delay without sacrificing much of the offloading efficiency. The performance of the scheme is mathematically analyzed using matrix-geometric techniques and validated through simulation.

Adaptive lead time quotation in a pull production system with lead time responsive demand

In this paper, a single-stage single-class kanban controlled manufacturing system is considered, where the problem is quoting accurate lead times for orders that are generated by an MRP system. Order release mechanism as in classical MRP is sensitive to the changes in the lead time according to the concept of lead time syndrome. The objective is to establish a cost effective lead time quoting procedure. The problem is modeled as a two-dimensional Markov chain, and it is solved explicitly by using matrix geometric techniques. Comparative analysis is done between static and dynamic lead time quoting procedures, and significant cost benefits of the dynamic procedure are shown under various scenarios. Guidelines for setting design parameters such as the number of kanbans and the frequency of updating the lead time are provided through numerical tests.

Performance modeling of QoS in a multicode multicarrier CDMA wireless network with fading

AbstractFor emerging wireless mesh networks, multicode multicarrier CDMA(MC‐CDMA) based technology is one of the most viable candidates. We perform stochastic modelling of the queues at the mobile station for uplink communication with multicode multicarrier CDMA system with two types of traffic, namely real‐time and non‐real‐time. Each traffic is assigned its own codes. However, the non‐real‐time traffic is allowed to use codes assigned to real‐time traffic, when real‐time traffic is not using its codes. Based on the probability of bit error for a multicode MC‐CDMA system, we first compute the probability of packet error. The packet in error will be inserted into the queue until it successfully gets through to the receiver. The packet arrival process at the input queue is modelled as Markov modulated Poisson processes (MMPP). The QoS performance in terms of packet loss for real‐time traffic and the occupancy distribution for non‐real‐time traffic is evaluated using matrix geometric techniques. We present numerical results for low and high load of real‐time traffic with varying loads of non‐real‐time traffic. We observe the binomial tweaking feature of occupancy distribution at higher loads due to batch departures. Copyright © 2009 John Wiley & Sons, Ltd.

A conditional probability approach to [formula omitted]/ [formula omitted]/1-like queues

Following up on a recently renewed interest in computational methods for M / G /1-type processes, this paper considers an M / G /1-like system in which the service time distribution is represented by a Coxian series of memoryless stages. We present a novel approach to the solution of such systems. Our method is based on conditional probabilities, and provides a simple, computationally efficient and stable approach to the evaluation of the steady-state queue length distribution. We provide a proof of the numerical stability of our method. Without explicit use of matrix-geometric techniques or stochastic complementation, we are able to handle systems with state-dependent service and arrival rates. The proposed approach can be used to compute the queue length distribution for both finite and infinite M / G /1-like queues. In the case of an infinite, state-independent queue, our method allows us to show using elementary tools that the queue length distribution is asymptotically geometric. The parameter of the asymptotic geometric can be expressed through a simple set of equations, easily solved using fixed point iteration. Our approach is very thrifty in terms of memory requirements, easy to implement, and generally fast. Numerical examples illustrate the performance of the proposed method.

Efficient steady-state analysis of second-order fluid stochastic Petri nets

This paper presents an efficient solution technique for the steady-state analysis of the second-order stochastic fluid model underlying a second-order fluid stochastic Petri net (FSPN) with constant flow and transition rates, and a single bounded fluid place. The new solution technique is an extension of existing solution techniques developed for (first-order) Fluid Models; the solution algorithm uses upwind semi-discretization and Matrix Geometric techniques to efficiently compute the steady-state probabilities of the mixed discrete and continuous state space of the model. The effectiveness of our technique is proven first on a simple producer–consumer second-order FSPN model and then on a complex example taken from the literature where the analysis of the completion time distribution and the packet loss probability of short-lived TCP connections is investigated through the decomposition of a model for the whole system into several (simpler) sub-models; the interaction between the different sub-models is handled by iterating their solution until the complete model solution converges according to a fixed point algorithm. The introduction of a second-order FSPN in a fixed point iteration scheme has been made possible thanks to the efficiency of the proposed solution technique.

Assessing the benefits of remanufacturing option under one-way substitution

In this study, we consider a segmented market for a product that can either be manufactured or remanufactured. It is assumed that the remanufactured products can be substituted by the new ones. A steady-state profit model is constructed under certain environmental assumptions on capacity requirements of operations, and revenue and cost schemes. Exact steady-state probabilities of the Markovian model constructed are solved via matrix geometric techniques. An extensive computational study is performed to investigate the conditions under which the utilization of remanufacturing option and the use of one-way substitution policy increase the average expected profit.

COMPUTATION OF STEADY-STATE PROBABILITIES FOR RESOURCE-SHARING CALL-CENTER QUEUEING SYSTEMS

Two routing rules for a queueing system of two stations are considered as alternative models for modeling a call-center network. These routing rules allow customers to switch queues under certain server and other resource availability conditions, either external to the system upon arrival to the network, or internal to the system after arrival to a primary call center. Under the assumption of Poisson arrivals and exponentially distributed service times, these systems are analyzed using matrix-geometric techniques, yielding a non-trivial set of ergodicity conditions and the steady-state joint probability distribution for the number of customers at each station. An extensive numerical analysis is conducted, yielding some physical insight into these systems and related generalizations.

Model and performance evaluation for multiservice network link supporting ABR and CBR services

We consider a "dynamic" model for a link jointly supporting two types of service: rate-adaptive and constant-bit rate services, e.g., ABR and CBR service in ATM networks. We construct and solve the associated two dimensional Markov chain via matrix-geometric techniques. This permits an analytical evaluation of performance when such services share resources. Moreover, when constant-bit rate connections are long lived relative to rate-adaptive connections (e.g., file transfers), we prove a separation of time scale result. This leads to a useful approximation that closely matches the performance in this regime.

Provable bounds for the mean queue lengths in a heterogeneous priority queue

We analyze a two-class two-server system with nonpreemptive heterogeneous priority structures. We use matrix–geometric techniques to determine the stationary queue length distributions. Numerical solution of the matrix–geometric model requires that the number of phases be truncated and it is shown how this affects the accuracy of the results. We then establish and prove upper and lower bounds for the mean queue lengths under the assumption that the classes have equal mean service times.

Queue Size Distribution of an ATM Switch with VBR Video Traffic

Many analytical tools have been devised to analyze CBR video coders, however only few have been applied to the VBR case. In this paper, we present a semi-Markov process model for approximating the stochastic equilibrium queue length distribution for VBR video sources multiplexed onto a constant capacity ATM channel. The sources are assumed to have two rates, and generate cells that pass through pre-buffers before joining a common queue on a FIFO basis. The model is based on a state process defined by the average rate of change of the queue length as a function of the number of active mini-sources of each type, and is solved using matrix-geometric techniques. An Access Control mechanism uses these results inside its policing function to control access to the network at call setup. Numerical results are obtained for various conditions, and compared to the ones obtained from simulation and fluid-flow models, and good agreement was achieved.

Approximate analysis of networks of PH|PH|1|K queues with customer losses: Test results

We address the approximate analysis of large, open networks of general finite-buffer queues. We start from the analysis of large, open infinite-buffer queueing networks (QNs), as proposed by Kuhn and later extended by Whitt, in which large QNs are decomposed into a number of individual GI|G|1 queues, characterized by the first and second moment of the service and interarrival time distribution. We restrict ourselves by focusing on PH|PH|1 queues for which matrix-geometric and general Markovian techniques can be used, and allow for the inclusion of finite-buffers, i.e., we address networks of PH|PH|1|K queues in which customers arriving at completely filled queues are lost. In doing so, the above decomposition can not be done in a single step but should be done iteratively, i.e., we propose a fixed-point scheme for the solution of networks of PH|PH|1|K queues. In this paper we focus on the accuracy of the fixed-point approach. To keep the current paper self-contained, we describe the theory as well. We have tested our approach on a large set of examples and compared the results with simulations. It turns out that our approach does not only yield accurate results (mostly within a few percents from simulation results) but is also very fast in obtaining them (in comparison with simulation). To the best of our knowledge, it is currently the only approach for analysing large networks of finite-buffer PH|PH|1|K queues.

PROFIT ANALYSIS OF THE M/E k/1 MACHINE REPAIR PROBLEM WITH A NON-RELIABLE SERVICE STATION

This paper studies the machine repair problem in which N identical automatic machines are maintained by a single non-reliable service station. Failure time and service time of the machines are assumed to follow the negative exponential distribution and the Erlang k type distribution, respectively. Breakdown time and repair time of the service station are also assumed to follow the negative exponential distribution and the Erlang m type distribution, respectively. Matrix geometric techniques are utilized to derive the steady-state probabilities of the number of failed machines in the system. A profit model is developed to determine the optimum number of machines to be assigned to the service station in order to maximize the total expected profit per machine per unit time. © 1997 Elsevier Science Ltd

Queueing analysis of ATM multicast switching models

The performance of a multicast switching system with a cyclic priority input access scheme is evaluated by means of an analytic model which may treat two &tinct multicast switches in a unified way. The performance measure is the packet delay at the input buffer of the system. The analysis is subsequently performed using matrix-geometric techniques and the numericai results are compared with simulations.

Performance study of ATM multicast switching systems

AbstractWe evaluate the performance of an N × N ATM discrete time multicast switch model with input queueing operating under two input access disciplines. First we present the analysis for the case of a purely random access discipline and subsequently we concentrate on a cyclic priority access based on a circulating token ring. In both cases, we focus on two HOL (head‐of‐line) packet service disciplines. Under the first (one‐shot transmission discipline), all the copies generated by each HOL packet seek simultaneous transmission during the same time slot. Under the second service discipline (call‐splitting), all HOL copies that can be transmitted in the same time slot are released while blocked copies compete for transmission in subsequent slots. In our analysis the performance measures introduced are the average packet delay in the input buffers as well as the maximum throughput of the switch. A significant part of the analysis is based on matrix geometric techniques. Finally, numerical results are presented and compared with computer simulations.

Access control of bursty voice traffic in ATM networks

Asynchronous Transfer Mode (ATM) Broadband Networks are expected to provide a large number of services and applications. A predominant service will be the support of multimedia traffic applications that integrate different types of traffic (e.g., video, voice and data) over the same connection. The heterogeneous traffic-types differ, widely, not only in their individual bit-rates but also in their corresponding correlations and burstiness parameters. Access flow control algorithms are required to enforce and maintain a specific QOS for each type of traffic. In this paper, we present an effective access flow control algorithm that can be applied at the user to network interface (UNI). The scheme is based upon a preventive control principle and uses a control signal, that is function of the buffer occupancy level, to control the source coder sampling rate. The scheme avoids congestion and enhances the statistical multiplexing gain, consequently providing a higher bandwidth utilization for each QOS. We provide a performance analysis study, supported by simulations, of the access node multiplexer. We studied the resulting Quasi Birth Death queueing process, with state dependant arrivals, using matrix-geometric techniques, and found that both MMPP/Er/1/ K and the Poisson Process approximations provided very close results for an ATM multiplexer with small buffer size and access control.

The difference of two renewal processes level crossing and the infimum

We consider the difference process N of two independent renewal (counting) processes. Second-order approximations to the distribution function of the level crossing time are given. Direct application of the second-order approximation is complicated by the occurrence of an (in general) unknown term E[Mtilde], which denotes the expected minimum of the stationary version of N. However, this number is obtained for a wide class of processes N, using matrix-geometric techniques. Numerical experiments have been carried out, in which the new approximations were compared to simulation, first-order and/or exact results. These results confirm that the second-order approximations are considerably better than the (known) first-order ones. We consider the difference process N of two independent renewal (counting) processes. Second-order approximations to the distribution function of the level crossing time are given. Direct application of the second-order approximation is complicated by the occurrence of an (in general) unknown term E[Mtilde], which denotes the expected minimum of the stationary version of N. However, this number is obtained for a wide class of processes N, using matrix-geometric techniques. Numerical experiments have been carried out, in which the new approximations were compared to simulation, first-order and/or exact results. These results confirm that the second-order approximations are considerably better than the (known) first-order ones.

A multiserver queue with narrow- and wide-band customers and wide-band restricted access

A multiserver queueing system with two classes of customers is considered: a type-1 (narrowband, NB) customer requires a single server, while each type-2 (wideband, WB) customer requests n of the m servers (n is not random). Severs allocated to a type-2 customer are seized and released simultaneously. Service times are exponentially distributed with mean 1//sub mu i/ for type i customers (i=1, 2). Blocked type-1 customers are cleared while blocked type-2 customers may be delayed in an infinite waiting room. A type-1 customer enters service immediately upon arrival if at least one server is free, irrespective of the status of the type-2 queue. WB customers have restricted access to the service facility; a cutoff parameter specifies the maximum number of type-2 customers that can be in service at the same time. Two approaches, moment-generating functions and matrix-geometric techniques, are considered for the computation of the system performance, that is, the mean waiting time in queue and the probability of delay (i.e. nonzero waiting time) for type-2 customers, as well as the probability of blocking for type-1 customers. >