On a tandem queueing model with identical service times at both counters, I

This paper studies tandem queues where the ith customer has the same service time, Xi, at each queue and capacity at each queue is finite. The behaviour of the tandem queue is examined in the presence and absence of blocking. For bounded service time distributions, it is already known that provided there is sufficient capacity available to prevent blocking, it is optimal to allocate the capacity uniformly. We show that for some simple service time distributions with support on two points, the throughput can be calculated exactly and that it is always optimal to allocate the capacity as uniformly as possible, even when blocking occurs. These results contrast with those of previous authors that suggest that a ‘reversed-bowl’ allocation of capacity may be optimal when service times are independent from queue to queue.

The aim of this paper is to analyze a tandem queueing model with two stages. The arrivals to the first stage are Poisson stream and the service time at this stage is exponential. There is no waiting room at first stage. The service time is hyperexponential and no waiting is allowed at second stage. The transition probabilities and loss probabilities of this model are obtained. In addition, the loss probability at second stage is optimized. Performance measures and the variance of the numbers of customers of this tandem queueing model are found. It is seen that the numbers of customers in first stage and second stage are dependent. Finally we have simulated this queueing model. For different values of parameters, exact values, simulated values, and optimal values of obtained performance measures of this model are numerically shown in tables and graphs.

The analysis of system performance measures is fairly simple if there is only one or several channels with identical service times and arranged in parallel with exponentially distributed inter arrival time of packets and service time. However, heterogeneous multiple channels often occur in practice and here the channels have different service times. It is therefore useful to be able to calculate the performance measures of such systems. We shall take an exact analytical model of the asymmetric data network node and assume that the arrival process is Poisson of two priority flows with different rates and two channels with different service times distributed exponentially and in case with determine service time. In symmetric queuing systems, the response time depends only on whether an arrival message finds some free channel. An asymmetric multiple channel system differs since which of the channels processes the message. It is very important to take analysis of different strategies for selecting a free channel. In a system with heavy traffic there is usually all channels are occupied and therefore a channel selection strategy is not so efficient. The transmission order is from high-priority packet to low-priority packet. Two priority packets queues share a limited size single buffer. The each priority packet is dropped if the queue length exceeds a capacity of buffer. Network node performance measures such as packet losses, packet delay, channel utilization are evaluated by means of analytical and simulation models. Some calculated system performance measures results are presented in figures.

Tandem queues are assumed to have several stations (stages, phases) in series. In such systems after receiving service in a certain station, a customer goes to the next station to get service. Tandem queues can be used for modeling real-life networks with a linear topology as well as for the validation of general decomposition algorithms in networks. Thus, tandem queueing systems have found much interest in the literature. Most of early papers on tandem queues are devoted to exponential queuing models. Over the last two decades efforts of many investigators in tandem queues were directed to weakening the distribution assumptions on the arrival pattern and service times. In particular, the arrival process should be able to capture correlation and explosive nature of real traffic in modern communication networks. As it was noted in Sect. 6.2, the correlation and the variability of inter-arrival times in input flows are taken into account via the consideration of a BMAP. In this section, we consider a number of two-station and multi-station tandem queues with BMAP, general or PH distribution of service times and different configuration of stations.

Tandem queues constitute a fundamental structure of queueing networks. However, exact queue times in tandem queues are notoriously difficult to compute except for some special cases. Several approximation schemes that are based on mathematical assumptions that enable approximate analyses of tandem queues have been reported in the literature. This article proposes an approximation approach that is based on observed properties of the behavior of tandem queues: the intrinsic gap and intrinsic ratio. The approach exploits the nearly linear and heavy-traffic properties of the intrinsic ratio, which appear to hold in realistic production situations. The proposed approach outperforms existing approximation methods across a broad range of examined cases. It is also demonstrated that the proposed approach has the potential when applied to historical data to achieve accurate mean queue time estimates in practical production environments. [Supplemental materials are available for this article. Go to the publisher's online edition of IIE Transactions to view the supplemental file.]

It is well known that queues with exponentially distributed service times have the smallest Shannon capacity among all single-server queues with the same service rate. In this paper, we study the capacity of timing channels in which the service time distributions have bounded support, i.e., Bounded Service Timing Channels (BSTC). We derive an upper bound and two lower bounds on the capacity of such timing channels. The tightness of these bounds is investigated analytically as well as via simulations. We find that the uniform BSTC serves a role for BSTCs that is similar to what the exponential service timing channel does for the case of timing channels with unbounded service time distributions. That is, when the length of the support interval is small, the uniform BSTC has the smallest capacity among all BSTCs.

New bounds for expected cycle times in tandem queues with blocking

In this paper, a single-server queue with non-homogeneous Poisson input and general service time is considered. Particular attention is given to the case where the parameter of the Poisson input λ(t) is a periodic function of the time. The approach is an extension of the work of Takács and Reich . The main result of the investigation is that under certain conditions on the distribution of the service time, the form of the function λ(t) and the distribution of the waiting time at t = 0, the probability of a server being idle P 0 and the Laplace transform Ω of the waiting time are both asymptotically periodic in t. Putting where b(t) is a periodic function of time, it is shown that both P o and Ω can be expanded in a power series in z, and a method for calculating explicitly the asymptotic values of the leading terms is obtained. In many practical queueing problems, it is expected that the probability of arrivals will vary periodically. For example, in restaurants or at servicestations arrivals are more probable at rush hours than at slack periods, and rush hours are repeated day after day

A model for the service time structure in the single server queue is given embodying correlations between contiguous and near-contiguous service times. A number of results are derived in the case of Poisson arrivals both for equilibrium and the transient state. In particular, Kendall's (equilibrium) resultP(a departure leaves the queue empty) = 1 — (mean service time)/(mean inter-arrival time) is found still to hold good. The effect of the correlation on the mean and variance of the equilibrium queue length distribution is examined in a simple case.

Due to the analytical intractability of general tandem queues, we develop methods to quantify the dependence among servers through simulation. Dependence is defined by the contribution queue time at each server, and contribution factors are developed based on the insight from Friedman's reduction method and Jackson networks. In a tandem queue, the dependence among stations can be either diffusion or blocking, and their impact depends on the positions relative to the bottlenecks.

Many transportation systems rely on information technology (IT) for effective and efficient system operations. When the IT systems or resources are shared among various processes or servers, the service times of these processes or servers become correlated. Failure to recognize such correlations in model development may lead to significant inaccuracies. The effects of various types of correlation in the context of border crossing operations were investigated. Several possible reasons for correlation among servers and the ways in which such correlations may affect system operations are explained. Three scenarios are investigated through numerical experimentation: correlated tandem queues, correlated parallel queues, and the impact of correlated parallel queues on a downstream service station. Models are developed with and without consideration for the correlation for these three cases. Ignoring correlation in the performance analysis (i.e., assuming that the servers are independent) leads to inaccurate results, especially when system use is great, that is, at high levels of congestion. In the case of parallel queues, independence assumption leads to overestimated system capacity and to underestimated capacity in tandem queues.

We establish a conservation law for G/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in a G/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in a GI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

Some statistical analysis of design data.- Line service times and utilization.- First concepts and relations of queuing theory.- A first resuot on the single-server queue.- A general result on the single-server queue.- Termina systems with polling.- Some results on queues with priority disciplines.- Some applications of Markov processes.- Core buffer requirements.- Service times of direct-access storage devices.- Queuing for data accesses.- Final remarks on the computer center analysis.- Imbedded Markov chains of some queuing processess.- Limit utilization and wait time distribution.

A tandem queue with a FIFO multiserver system at each stage, i.i.d. service times and a renewal process of external arrivals is shown to be regenerative by modeling it as a Harris-ergodic Markov chain. In addition, some explicit regeneration points are found. This generalizes the results of Nummelin (1981) in which a single server system is at each stage and the result of Charlot et al. (1978) in which the FIFO GI/GI/c queue is modeled as a Harris chain. In preparing for our result, we study the random assignment queue and use it to give a new proof of Harris ergodicity of the FIFO queue.

This study analyzed a tandem queueing system with blocking and no waiting line. The arrival process is assumed to be Poisson with rate. There are two non-identical servers in the system. The service times are exponentially distributed with parameters and at servers 1 and 2, respectively. Besides, the catastrophes occur in a Poisson manner with rate in the system. When server 1 is busy or blocked, the customer who arrives in the system leaves the system without being served. Such customers are called “lost customers”. The probability of losing a customer was computed for the system. Key words: Tandem queueing, heterogeneous servers, catastrophes, loss probability.