Sojourn time analysis for a cyclic-service tandem queueing model with general decrementing service

A sojourn time analysis is provided for a cyclic-service tandem queue with general decrementing service which operates as follows: starting once a service of queue 1 in the first stage, a single server continues serving messages in queue 1 until either queue 1 becomes empty, or the number of messages decreases to k less than that found upon the server's last arrival at queue 1, whichever occurs first, where 1 ≤ k ≤ ∞. After service completion in queue 1, the server switches over to queue 2 in the second stage and serves all messages in queue 2 until it becomes empty. It is assumed that an arrival stream is Poissonian, message service times at each stage are generally distributed and switch-over times are zero. This paper analyzes joint queue-length distributions and message sojourn time distributions.     

On first passage time structure of random walks

For continuous time birth-death processes on {0,1,2,…}, the first passage time T⁺_n from n to n + 1 is always a mixture of (n + 1) independent exponential random variables. Furthermore, the first passage time T_0,n+1 from 0 to (n + 1) is always a sum of (n + 1) independent exponential random variables. The discrete time analogue, however, does not necessarily hold in spite of structural similarities. In this paper, some necessary and sufficient conditions are established under which T⁺_n and T_0,n+1 for discrete time birth-death chains become a mixture and a sum, respectively, of (n + 1) independent geometric random variables on {1,2,…};. The results are further extended to conditional first passage times.     

Sojourn times in queues with instantaneous tri-route decision process

In this paper, we study the total sojourn time in a queueing system with an instantaneous tri-route decision process. Even though the computations are more difficult, we give here the structure of the sojourn time process for the M/G/1 queue with tri-route decision process. A numerical study is carried out in this paper.     

Sojourn time distributions in the queue defined by a general QBD process

We consider a general QBD process as defining a FIFO queue and obtain the stationary distribution of the sojourn time of a customer in that queue as a matrix exponential distribution, which is identical to a phase-type distribution under a certain condition. Since QBD processes include many queueing models where the arrival and service process are dependent, these results form a substantial generalization of analogous results reported in the literature for queues such as the PH/PH/c queue. We also discuss asymptotic properties of the sojourn time distribution through its matrix exponential form.     

Departures from queues with changeover times

We consider the M/M ^ij /1 queue as a model of queues with changeover times, i.e., the service is exponential with parameter _ij depending on the previous job type (i) and the current job type (j). It is shown that the departure process is renewal and Poisson iff _ij = (constant). In this case, types of departures are dependent renewal processes. Crosscovariance and crosscorrelations are given.     

Sojourn time distribution in a MAP/M/1 processor-sharing queue

This paper considers the sojourn time distribution in a processor-sharing queue with a Markovian arrival process and exponential service times. We show a recursive formula to compute the complementary distribution of the sojourn time in steady state. The formula is simple and numerically feasible, and enables us to control the absolute error in numerical results. Further, we discuss the impact of the arrival process on the sojourn time distribution through some numerical examples.     

Sojourn times with finite time-horizon in finite semi-markov processes

Let Y = {Y_t:t ≥ 0} be a semi-Markov process with finite state space S. Assume that Y is either irreducible and S is then partitioned into two classes A and B, or, that Y is absorbing and S is partitioned into A, B and C, where C is the set of all absorbing states of Y. Denote by T_{A, m}(t) the mth sojourn time of Y in A during [0, t]. T_{A, m}(t) is thus defined as the duration in [0, t] of the mth visit of Y to A if A is visited by Y during [0, t] at least m times; T_{A, m}(t) = 0 otherwise. We derive a recurrence relation for the vectors of double Laplace transforms g_m**(T₁,T₂) = {g_m**(T₁, T₂;S):sSC}, m = 1,2,… which are defined by with T₁, T₂, Re(T₁), Re(T₂) > 0. This result is then applied to alternating renewal processes. Symbolic Laplace transform inversion with MAPLE is used to obtain the first two moments of T_{A, m}(t). The assumed holding time distributions are exponential and Erlang respectively. This paper is a continuation of some of the author's recent work on the distribution theory of sojourn times in a subset of the finite state space of a (semi-)Markov process where the time horizon t = + ∞. The practical importance of considering a finite time horizon for semi-Markov reliability models has been discussed recently by Jack (1991), Jack and Dagpunar (1992), and Christer and Jack (1991).     

On performance comparison of MR/GI/1 queues

In this paper we are interested in the effect that dependencies in the arrival process to a queue have on queueing properties such as mean queue length and mean waiting time. We start with a review of the well known relations used to compare random variables and random vectors, e.g., stochastic orderings, stochastic increasing convexity, and strong stochastic increasing concavity. These relations and others are used to compare interarrival times in Markov renewal processes first in the case where the interarrival time distributions depend only on the current state in the underlying Markov chain and then in the general case where these interarrivai times depend on both the current state and the next state in that chain. These results are used to study a problem previously considered by Patuwo et al. [14].Then, in order to keep the marginal distributions of the interarrivai times constant, we build a particular transition matrix for the underlying Markov chain depending on a single parameter,p. This Markov renewal process is used in the Patuwo et al. [14] problem so as to investigate the behavior of the mean queue length and mean waiting time on a correlation measure depending only onp. As constructed, the interarrival time distributions do not depend onp so that the effects we find depend only on correlation in the arrival process.As a result of this latter construction, we find that the mean queue length is always larger in the case where correlations are non-zero than they are in the more usual case of renewal arrivals (i.e., where the correlations are zero). The implications of our results are clear.     

A BMAP/SM/1 queue with service times depending on the arrival process

We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival phase states, that is, there are many types of arrivals which have different service time distributions. The service process is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained. The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship between the stationary waiting time distribution and the stationary distribution of the number of customers in the system at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types of customers at departures. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some first passage time problems for the shortest queue model

We consider the symmetric shortest queue (SQ) problem. Here we have a Poisson arrival stream of rate λ feeding two parallel queues, each having an exponential server that works at rate μ. An arrival joins the shorter of the two queues; if both are of equal length the arrival joins either with probability 1/2. We consider the first passage time until one of the queues reaches the value m ₀, and also the time until both reach this level. We give explicit expressions for the first two first passage moments, conditioned on the initial queue lengths, and also the full first passage distribution. We also give some asymptotic results for m ₀→∞ and various values of ρ=λ/μ. H. Yao work was partially supported by PSC-CUNY Research Award 68751-0037. C. Knessl work was supported in part by NSF grants DMS 02-02815 and DMS 05-03745.     

Some algorithms for discrete time queues with finite capacity

We consider a discrete time queue with finite capacity and i.i.d. and Markov modulated arrivals. Efficient algorithms are developed to calculate the moments and the distributions of the first time to overflow and the regeneration length. Results are extended to the multiserver queue. Some illustrative numerical examples are provided. This revised version was published online in June 2006 with corrections to the Cover Date.     

A note on the hypercube model

We consider a variation of the hypercube model in which there are N distinguishable servers and R types of customers. Customers that find all servers busy (blocked customers) are lost. When service times are exponentially distributed and customers arrive according to independent Poisson streams, we show that the policy which always assigns customers to the fastest available server minimizes the long-run average number of lost customers. Furthermore, we derive an upper bound for the blocking probability and the long-run average number of customers lost.     

Sojourn time asymptotics in processor-sharing queues

Over the past few decades, the Processor-Sharing (PS) discipline has attracted a great deal of attention in the queueing literature. While the PS paradigm emerged in the sixties as an idealization of round-robin scheduling in time-shared computer systems, it has recently captured renewed interest as a useful concept for modeling the flow-level performance of bandwidth-sharing protocols in communication networks. In contrast to the simple geometric queue length distribution, the sojourn time lacks such a nice closed-form characterization, even for exponential service requirements. In case of heavy-tailed service requirements however, there exists a simple asymptotic equivalence between the sojourn time and the service requirement distribution, which is commonly referred to as a reduced service rate approximation. In the present survey paper, we give an overview of several methods that have been developed to obtain such an asymptotic equivalence under various distributional assumptions. We outline the differences and similarities between the various approaches, discuss some connections, and present necessary and sufficient conditions for an asymptotic equivalence to hold. We also consider the generalization of the reduced service rate approximation to several extensions of the M/G/1 PS queue. In addition, we identify a relationship between the reduced service rate approximation and a queue length distribution with a geometrically decaying tail, and extend it to so-called bandwidth-sharing networks. The state-of-the-art with regard to sojourn time asymptotics in PS queues with light-tailed service requirements is also briefly described. Last, we reflect on some possible avenues for further research. AMS Subject Classification 60K25 (primary), 60F10, 68M20, 90B18, 90B22 (secondary).     

Bulk input queues with hysteretic control

This paper deals with a bulk input-batch service queueing system and (r,N)-hysteretic control, i.e., under N-policy and r-quorum discipline. The model also includes state dependent service. The goal of such global control is to reduce the waste of server capacity (r-quorum), an unwanted number of switchovers between idle and busy modes (N-policy), and the queue length (by means of variable service rates). The analysis of the system is based on first excess level technique developed by the second author. This approach enables the authors to obtain major characteristics for the queueing process in a closed analytical form. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analysis of discrete time queues via the reversed process

A class of discrete-timeM/G/1 queues, including both round robin and last come first served service, in which customers are subject to permutations is considered. These time slotted queues, analogous to the symmetric queues of Kelly, are analyzed by examination of the time reversed process. Product form stationary distributions are found for a type of doubly stochastic server of Schassberger [5] and for a Bernoulli arrival process queue model of Henderson and Taylor [2].     

Sojourn time analysis of a two-phase queueing system with exhaustive batch-service and its vacation model

We consider an M/G/1-type, two-phase queueing system, in which the two phases in series are attended alternatively and exhaustively by a moving single-server according to a batch-service in the first phase and an individual service in the second phase. We show that the two-phase queueing system reduces to a new type of single-vacation model with non-exhaustive service. Using a double transform for the joint distribution of the queue length in each phase and the remaining service time, we derive Laplace-Stieltjes transforms for the sojourn time in each phase and the total sojourn time in the system. Furthermore, we provide the moment formula of sojourn times and numerical examples of an approximate density function of the total sojourn time.     

Information theory and the failure time of a system

In this paper we introduce a measure for the rate of generation of information about the failure time of a system using mutual information measure. In the case of a system with multi-components what is really measured is the interaction between one component of the system and the rest. Our definition of measure is only a slight variation of existing classical definitions in the information theory literature. We study properties of our proposed measure and calculate information for several hypothetical systems.This research was partially supported by the U.S. Air Force Office of Scientific Research Grant AFSOR-89-0402.     

Busy-period and blocking behavior of finite queues with state-dependent Markov renewal arrivals

The busy-period length distributions and blocking probabilities are considered for finiteG/G/1/K queues with state-dependent Markov renewal arrivals. The Laplace-Stieltjes transforms of the distributions and blocking probabilities are given for the non-preemptive and last-come-first-served preemptive resume (or repeat) service disciplines. For Erlangian (or deterministic) service times in particular, it is proved that the busy-period length (the number of blocked customers) for the non-preemptive discipline is smaller (larger) than for the preemptive resume discipline.     

Monotone matrices and monotone Markov processes

A stochastic matrix is “monotone” [4] if its row-vectors are stochastically increasing. Closure properties, characterizations and the availability of a second maximal eigenvalue are developed. Such monotonicity is present in a variety of processes in discrete and continous time. In particular, birth-death processes are monotone. Conditions for the sequential monotonicity of a process are given and related inequalities presented.     

Kitting process in a stochastic assembly system

In small-lot, multi-product, multi-level assembly systems, kitting (or accumulating) components required for assembly plays a crucial role in determining system performance, especially when the system operates in a stochastic environment. This paper analyzes the kitting process of a stochastic assembly system, treating it as an assembly-like queue. If components arrive according to Poisson processes, we show that the output stream departing the kitting operation is a Markov renewal process. The distribution of time between kit completions is also derived. Under the special condition of identical component arrival streams having the same Poisson parameter, we show that the output stream of kits approximates a Poisson process with parameter equal to that of the input stream. This approximately decouples assembly from kitting, allowing the assembly operation to be analyzed separately.