Manuel Alberto M. Ferreira
In this article it is shown that if the busy period of a MGinf queue system is PME distributed, the service time is a random variable with a long tail distribution. The result is obtained through Laplace transforms analysis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Solving a Riccati equation, induced by the study of the transient behaviour of the MGInf queue system, a collection of service times distributions is determined. For the MGInf queue, which service time distribution is a member of that collection, the busy period and busy cycle probabilistic studies are performed. In extra, the properties of that distributions collection are deduced and presented.
We present formulas to compute the busy cycle renewal function for the $M|G|\infty$ queue and exemplify for some service time distributions. The busy cycle renewal function value in t is the number of busy periods that begin in the interval from 0 till $t$. This number is of crucial importance in practical applications of $M|G|\infty$ queue.
The busy period length distribution function knowledge is important for any queue system, and for the MGINF queue. But the mathematical expressions are in general very complicated, with a few exceptions, involving usually infinite sums and multiple convolutions. So, in this work are deduced some bounds for the MMINF system busy period length distribution function, meaning the second M exponential service time, which analytic expressions are simpler than the exact one. As a consequence, also some bounds for the MMINF system busy cycle length distribution function are presented.
In this paper we analyze the distributional properties of a busy period in an on-off fluid queue and the a first passage time in a fluid queue driven by a finite state Markov process. In particular, we show that in Anick-Mitra-Sondhi model the first passage time has a IFR distribution and the busy period has a DFR distribution.
In the infinite servers queue with Poisson arrivals real life practical applications, the busy period and the busy cycle probabilistic study is of main importance. But it is a very difficult task. In this text, we show that by solving a Riccati equation induced by this queue transient probabilities monotony study as time functions, we obtain a collection of service length distribution functions, for which both the busy period and the busy cycle have lengths with quite simple distributions, generally given in terms of exponential distributions and the degenerate at the origin distribution.
In this work it is proposed a parameter, η, to characterize the MGinf Queue Busy Period distribution that is a modification of the peakedness proposed in Whitt (1984). As this distribution moments centered at the origin are given, in general, by very complex expressions it is relevant to search for parameters simpler but with good descriptive properties.
The queue system,with Poisson arrivals,constant service time and infinite servers, busy period distribution is intensively studied because, due to its probability density function quite easy interpretation, it may serve as a clue to interpret the queue systems, with Poisson arrivals, infinite servers, and other service time distributions, busy period distributions. In this work the queue system, with Poisson arrivals,constant service time and infinite servers, busy period distribution parameters are exactly computed and its probability density function is interpreted and explained. The only pr…
Some important results on the variance of the $M|G|\infty$ queue busy period are presented. Often, this parameter depends on the whole structure of the service time distribution. So, the importance of the bounds presented, depending only on some parameters. Also, some bounds for service time distributions important in reliability theory, with technological and financial applications, are presented and a particular stochastic ordination is explored to explain those results. Finally, an interesting result about the exponentiality of the MGInf queue busy period, resulting from the coefficient of …
This paper purpose is to investigate exponential behavior conditions for the infinite servers queue with Poisson arrivals busy period length distribution. It is presented a general theoretical result that is the basis of this work. The complementary analysis rely on the infinite servers queue with Poisson arrivals busy period length distribution moments computation. In infinite servers queue with Poisson arrivals practical applications, in economic, management, and business areas, the management of the effective number of servers is essential since the physical presence of infinite servers is …
We discuss weak convergence of the number of busy servers in a $G/G/\infty$ queue in the $J_1$-topology on the Skorokhod space. We prove two functional limit theorems, with random and nonrandom centering, respectively, thereby solving two open problems stated in Mikosch and Resnick (2006}. A new integral representation for the limit Gaussian process is given.
We give a simple derivation of the distribution of the maximum L of the length of the queue during a busy period for the M/M/1 queue with lambda<1 the ratio between arrival rate and service rate. We observe that the asymptotic behavior of the moments of L is related to that of Lambert series for the generating functions for the sums of powers of divisors of positive integers. We show how to obtain asymptotic expansions for these moments with error terms having order as large a power of 1-lambda as desired.
The Laplace transform is a widely used tool in the study of probability distributions, often allowing for a probability density functions and distribution functions simpler determination and being a moments generating function. In this paper it is considered a situation not so simple, as it is the case of the MGinf queue busy period length distribution. Attention will also be given the respective tail Laplace transform. Then, in the context of an open queues network, which nodes behave as MGinf queues, the Laplace transform will be used to construct an algorithm to determine the Laplace transf…
We introduce a multiclass single-server queueing system in which the arrival rates depend on the current job in service. The system is characterized by a matrix of arrival rates in lieu of a vector of arrival rates. Our proposed model departs from existing state-dependent queueing models in which the parameters depend primarily on the number of jobs in the system rather than on the job in service. We formulate the queueing model and its corresponding fluid model and proceed to obtain the necessary and sufficient conditions for stability via fluid models. Utilizing the natural connection with t…
We study critical GI/G/1 queues under finite second moment assumptions. We show that the busy period distribution is regularly varying with index half. We also review previously known M/G/1/ and M/M/1 derivations, yielding exact asymptotics as well as a similar derivation for GI/M/1. The busy period asymptotics determine the growth rate of moments of the renewal process counting busy cycles. We further use this to demonstrate a BRAVO phenomenon (Balancing Reduces Asymptotic Variance of Outputs) for the work-output process (namely the busy-time). This yields new insight on the BRAVO effect. A s…
In this work, approximate and exact results concerning a performance measure of the MGinf system, the age or the excess of the busy cycle, are presented. It will be seen that it is also a measure of the busy period performance. Service distributions for which it is given for a simple expression and others for which this does not happen are considered. For this last case, bounds are deduced. A special emphasis is given to the exponential distribution and to those related with it, useful in reliability theory.
M|D|$\infty$ Queue Busy Period and Busy Cycle Distributions Computational Calculus
针对M|G|∞排队系统无忙期、忙循环分布闭式公式的问题，利用M|D|∞排队系统二者存在拉普拉斯闭式变换的特性，借助Platzman等人的算法，通过FORTRAN程序实现其分布函数计算。
Exact Results for the Distribution of the Partial Busy Period for a Multi-Server Queue
本文针对M/M/c多服务器队列，推导了部分忙期分布的精确显式结果，提出了谱方法和代数方法，并给出了大服务器数量下的渐近形式，是首个提供服务器数大于二时精确结果的工作。
We study the impact of service-time distributions on the distribution of the maximum queue length during a busy period for the M^X/G/1 queue. The maximum queue length is an important random variable to understand when designing the buffer size for finite buffer (M/G/1/n) systems. We show the somewhat surprising result that for three variations of the preemptive LCFS discipline, the maximum queue length during a busy period is smaller when service times are more variable (in the convex sense).
In this paper, we consider systems that can be modelled by $M \mid M \mid n$ queues with heterogeneous servers and non informed customers. Considering any two servers: we show that the probability that the fastest server is busy is smaller than the probability that the slowest server is busy. Moreover, we show that the effective rate of service done by the fastest server is larger than effective rate of service done by the slowest server.
The Gated-Service M/GI/1 Queue with Single Vacations and Its Application to Batch-Service Queues
本文研究了具有单个休假的Gated服务M/GI/1队列，推导了稳态队列长度的概率生成函数、系统延迟分布的Laplace-Stieltjes变换以及忙周期长度和服务顾客数的联合变换。同时，将结果应用于批量服务M/G/1队列，其中服务时间依赖于批量中的顾客数。
The infinite-server queueing models with homogeneous and non-homogeneous arrivals of customers and catastrophes are considered. The probability generating functions of joint distributions of numbers of busy servers and served customers, as well as the Laplace-Stieltjes Transforms of distribution of busy period and distribution of busy cycle for the models are found.
The problems arising when the moments of service time distributions, for which the MGinf queue system busy period and busy cycle become very easy to study, are presented and it is shown how to overcome them. The busy cycle renewal function and the peakedness and the modified peakedness for the MGinf busy period and busy cycle in the case of those service time distributions are also computed.
In this paper, we prove a characterization theorem on the number of losses during a busy period in $GI^X/GI^Y/1/n$ queueing systems, in which interarrival time distribution belongs to the class NWUE.
This paper considers a multichannel preemptive-resume priority queueing system with a Poisson input and an arbitrary service time distribution depending on the priority of job. Jobs of the same priority are serviced according to the LIFO rule. If, at moment of job arrival, all servers are busy, and at least one server is busy with the service of a job of a not higher than the priority of arriving job, then the service of a job is preempted such that the priority of preempted job is lowest from the priorities of the jobs in service. The service of a preempted job is resumed later. The paper pro…
We consider a horizontal traffic queue (HTQ) on a periodic road segment, where vehicles arrive according to a spatio-temporal Poisson process, and depart after traveling a distance that is sampled independently and identically from a spatial distribution. When inside the queue, the speed of a vehicle is proportional to a power $m >0$ of the distance to the vehicle in front. The service rate of HTQ is equal to the sum of the speeds of the vehicles, and has a complex dependency on the state (vehicle locations) of the system. We show that the service-rate increases (resp., decreases) in between a…
Fundamental Limits of Covert queueing under Busy-Period Observations
该研究针对忙期观测下的排队系统，证明Bob隐蔽服务额外作业的最大期望数量服从平方根定律，刻画了两类忙期统计量对应的隐蔽排队常数并对比其大小。
The busy period for a queue is cast as the area swept under the random walk until it first returns to zero, $B$. Encompassing non-i.i.d. increments, the large-deviations asymptotics of $B$ is addressed, under the assumption that the increments satisfy standard conditions, including a negative drift. The main conclusions provide insight on the probability of a large busy period, and the manner in which this occurs: I) The scaled probability of a large busy period has the asymptote, for any $b>0$, \lim_{n\to\infty} \frac{1}{\sqrt{n}} \log P(B\geq bn) = -K\sqrt{b}, \hbox{where} \quad K = 2 \sqrt{…
The interest to retrial queueing systems is due to their application to telephone systems. The paper studies multiserver retrial queueing systems with $n$ servers. Arrival process is a quite general point process. An arriving customer occupies one of free servers. If upon arrival all servers are busy, then the customer waits for his service in orbit, and after random time retries more and more to occupy a server. The orbit has one waiting space only, and arriving customer, who finds all servers busy and the waiting space occupied, losses from the system. Time intervals between possible retrial…
We introduce a new formalism for dealing with networks of queues. The formalism is based on the Doi-Peliti second quantization method for reaction diffusion systems. As a demonstration of the method's utility we compute perturbatively the different time busy-busy correlations between two servers in a Jackson network.
We introduce a fractional generalization of the Erlang Queues $M/E_k/1$. Such process is obtained through a time-change via inverse stable subordinator of the classical queue process. We first exploit the (fractional) Kolmogorov forward equation for such process, then we use such equation to obtain an interpretation of this process in the queuing theory context. Then we also exploit the transient state probabilities and some features of this fractional queue model, such as the mean queue length, the distribution of the busy periods and some conditional distributions of the waiting times. Final…
In this paper, we consider a Lévy-driven fluid queueing system where the server may subject to breakdowns and repairs. In addition, the server will leave for a vacation each time when he finds an empty system. We cast the queueing process as a Lévy process modified to have random jumps at two classes of stopping times. By using the Kella-Whitt martingale method, we obtain the limiting distribution of the virtual waiting time process. Moreover, we investigate the busy period, the correlation structure and the stochastic decomposition properties. These results may be generalized to Lévy processe…
On Erlang Queue with Multiple Arrivals and its Time-changed Variant
本文提出了一种允许多次到达的Erlang排队模型，推导了其状态概率、平均队列长度等分布特性，并进一步研究了其经逆稳定从属子变换的时间变体的相关性能指标。
In this paper we analyze a single server queue with batch arrivals and semi-Markovian service times. We also include the feature that the first service of each busy period might have a different distribution than subsequent service times. Our generating function based approach allows us to determine the heavy traffic limit of the scaled queue-length distribution. It turns out that this distribution converges to an exponential distribution. Nonsurprisingly, the exceptional first service does not influence this limiting distribution. We identify a sufficient and necessary condition under which t…
The infinite servers queue with Poisson arrivals state transient probabilities, considering the time origin at the beginning of a busy period, mean and variance monotony as time functions is studied. These studies, for which results it is determinant the hazard rate function service time length, induce the consideration of two differential equations, one related with the mean monotony study and another with the variance monotony study, which solutions lead to some particular service time distributions, for which those parameters present specific behaviors as time functions.
Tempered Erlang Queue with Multiple Arrivals
针对多重到达 Erlang 排队系统，本文利用缓增稳定从属过程的首次击中时间进行时间变换，建立并求解分数阶差分-微分方程，给出状态概率、等价阶段表示、平均队长及多种关键时间分布。
We consider a single-server GI/GI/1 queueing system with feedback. We assume the service times distribution to be (intermediate) regularly varying. We find the tail asymptotics for a customer's sojourn time in two regimes: the customer arrives in an empty system, and the customer arrives in the system in the stationary regime. In particular, in the case of Poisson input we use the branching processes structure and provide more precise formulae. As auxiliary results, we find the tail asymptotics for the busy period distribution in a single-server queue with an intermediate varying service times…
Heavy-tail asymptotics for the length of a busy period in a Generalised Jackson Network
研究稳定 Generalised Jackson Network 中忙期长度的重尾渐近性，证明其大值主要由单个异常大服务时间引起，并获得了 P(B>x) 的精确渐近表达式。
The areas under workload process and under queuing process in a single server queue over the busy period have many applications not only in queuing theory but also in risk theory or percolation theory. We focus here on the tail behaviour of distribution of these two integrals. We present various open problems and conjectures, which are supported by partial results for some special cases.
This work introduces the join the shortest queue policy in the retrial setting. We consider a Markovian single server retrial system with two infinite capacity orbits. An arriving job finding the server busy, it is forwarded to the least loaded orbit. Otherwise, it is forwarded to an orbit randomly. Orbiting jobs of either type retry to access the server independently. We investigate the stability condition, the stationary tail decay rate, and obtain the equilibrium distribution by using the compensation method.
After a short report of results on infinite servers queues systems, focusing on its busy period, using networks of queues with infinite servers nodes a model is constructed to study a two echelons repair system. These repair systems may be useful, for instance, in the operation of a fleet of aircraft, of shipping or of trucks. Additionally, it is shown how this model may be used in the evaluation of differential costs.
We study a queueing system with Erlang arrivals with $k$ phases and Erlang service with $m$ phases. Transition rates among phases vary periodically with time. For these systems, we derive the asymptotic periodic distribution of the level and phase as a function of time within the period. The asymptotic periodic distribution is analogous to a steady-state distribution for a system with constant rates. If the time within the period is considered part of the state, then it is a steady-state distribution. We also obtain waiting time and busy period distributions. These solutions are expressed as i…
The paper studies a multiserver retrial queueing system with $m$ servers. Arrival process is a point process with strictly stationary and ergodic increments. A customer arriving to the system occupies one of the free servers. If upon arrival all servers are busy, then the customer goes to the secondary queue, orbit, and after some random time retries more and more to occupy a server. A service time of each customer is exponentially distributed random variable with parameter $μ_1$. A time between retrials is exponentially distributed with parameter $μ_2$ for each customer. Using a martingale ap…
We revisit a single-server retrial queue with two independent Poisson streams (corresponding to two types of customers) and two orbits. The size of each orbit is infinite. The exponential server (with a rate independent of the type of customers) can hold at most one customer at a time and there is no waiting room. Upon arrival, if a type $i$ customer $(i=1,2)$ finds a busy server, it will join the type $i$ orbit. After an exponential time with a constant (retrial) rate $μ_i$, an type $i$ customer attempts to get service. This model has been recently studied by Avrachenkov, Nain and Yechiali~\c…
We investigate an M/M/1 queue operating in two switching environments, where the switch is governed by a two-state time-homogeneous Markov chain. This model allows to describe a system that is subject to regular operating phases alternating with anomalous working phases or random repairing periods. We first obtain the steady-state distribution of the process in terms of a generalized mixture of two geometric distributions. In the special case when only one kind of switch is allowed, we analyze the transient distribution, and investigate the busy period problem. The analysis is also performed b…
We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a certain Markov process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is always singular with respect to the Lebesgue measure which in many applications is ``close'' to reality. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of…
In the present paper the infinite-server MMAPkGk queueing model with random resource vector of customers, marked MAP arrival and semi-Markov (SM) arrival of catastrophes is considered. The joint generating functions (PGF) of transient and stationary distributions of number of busy servers and numbers of different types served customers, as well as Laplace transformations (LT) of joint distributions of total accumulated resources in the model at moment and total accumulated resources of served customers during time interval are found. The basic differential and renewal equations for transient a…
This paper considers the steady-state performance of load balancing algorithms in a many-server system with distributed queues. The system has $N$ servers, and each server maintains a local queue with buffer size $b-1,$ i.e. a server can hold at most one job in service and $b-1$ jobs in the queue. Jobs in the same queue are served according to the first-in-first-out (FIFO) order. The system is operated in a heavy-traffic regime such that the workload per server is $λ= 1 - N^{-α}$ for $0.5\leq α<1.$ We identify a set of algorithms such that the steady-state queues have the following universal s…
Suppose customers need to choose when to arrive to a congested queue with some desired service at the end, provided by a single server that operates only during a certain time interval. We study a model where the customers incur not only congestion (waiting) costs but also penalties for their index of arrival. Arriving before other customers is desirable when the value of service decreases with every admitted customer. This may be the case for example when arriving at a concert or a bus with unmarked seats or going to lunch in a busy cafeteria. We provide game theoretic analysis of such queuei…
We simulate queues of activity in a directed sandpile automaton in 1+1 dimensions by adding grains at the top row with driving rate $0 < r \leq 1$. The duration of elementary avalanches is exactly described by the distribution $P_1(t) \sim t^{-3/2}\exp{(-1/L_c)}$, limited either by the system size or by dissipation at defects $L_c= \min (L,ξ)$. Recognizing the probability $P_1$ as a distribution of service time of jobs arriving at a server with frequency $r$, the model represents a new example of the $<E,1,GI/\infty/1> $ server queue in the queue theory. We study numerically and analytically t…
We consider a polling system with two queues, exhaustive service, no switch-over times and exponential service times. The waiting cost depends on the position of the queue relative to the server: It costs a customer c per time unit to wait in the busy queue (where the server is) and d per time unit in the idle queue (where no server is). Customers arrive according to a Poisson process. We study the control problem of how arrivals should be routed to the two queues in order to minimize expected waiting costs and characterize individually and socially optimal routing policies under three scenari…
Steady-State Analysis and Online Learning for Queues with Hawkes Arrivals
本文研究具有霍克斯到达的单服务器队列的稳态行为，利用耦合技术建立矩界和指数收敛，并开发数据驱动的数值算法解决最优人员配置问题，发现与经典模型在重流量下存在显著差异。
We analyze an M/M/1 queue with a service discipline in which customers, upon arriving when the server is busy, search a sequence of stations for a vacant station at which to wait, and in which the server, upon becoming free when one or more customers are waiting, searches the stations in the same order for a station occupied by a customer to serve. We show how to find complete asymptotic expansions for all the moments of the waiting time in the heavy traffic limit. We show in particular that the variance of the waiting time for this discipline is more similar to that of last-come-first-served …
In this paper we study a storage process or a liquid queue in which the input process is the local time of a positively recurrent stationary diffusion in stationary state and the potential output takes place with a constant deterministic rate. For this storage process we find its stationary distribution and compute the joint distribution of the starting and ending times of the busy and idle periods. This work completes and extends to a more general setting the results in Mannersalo, Norros, and Salminen (2003).
In this paper, equilibrium strategies and optimal balking strategies of customers in a constant retrial queue with multiple vacations and the $N$-policy under two information levels, respectively, are investigated. We assume that there is no waiting area in front of the server and an arriving customer is served immediately if the server is idle; otherwise (the server is either busy or on a vacation) it has to leave the system to join a virtual retrial orbit waiting for retrials according to the FCFS rules. After a service completion, if the system is not empty, the server becomes idle, availab…
Queues with setup time are extensively studied because they have application in performance evaluation of power-saving data centers. In a data center, there are a huge number of servers which consume a large amount of energy. In the current technology, an idle server still consumes about 60\% of its peak processing a job. Thus, the only way to save energy is to turn off servers which are not processing a job. However, when there are some waiting jobs, we have to turn on the OFF servers. A server needs some setup time to be active during which it consumes energy but cannot process a job. Theref…
Age of Job Completion Minimization with Stable Queues
本文提出了一种基于马尔可夫机的作业调度策略，旨在通过最小化作业完成年龄和采样成本来优化作业完成效率。
We study stochastic tree fluid networks driven by a multidimensional Levy process. We are interested in (the joint distribution of) the steady-state content in each of the buffers, the busy periods, and the idle periods. To investigate these fluid networks, we relate the above three quantities to fluctuations of the input Levy process by solving a multidimensional Skorokhod reflection problem. This leads to the analysis of the distribution of the componentwise maximums, the corresponding epochs at which they are attained, and the beginning of the first last-passage excursion. Using the notion …
When decomposing the total orbit into $N$ sub-orbits (or simply orbits) related to each of $N$ servers and through comparing the numbers of customers in these orbits, we introduce a retrial supermarket model of $N$ identical servers, where two probing-server choice numbers are respectively designed for dynamically allocating each primary arrival and each retrial arrival into these orbits when the chosen servers are all busy. Note that the designed purpose of the two choice numbers can effectively improve performance measures of this retrial supermarket model. This paper analyzes a simple and b…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。