Analysis of two parallel queues by common exit service and infinite queue size

In Part I, a single-channel service system consisting of a service facility and queue, in which the possible times of arrival of units and the possible service times are discrete, is analyzed. A method of calculating the moments of the total service time of units in the system is developed. This total service time is related to the delay caused by the system. When arrivals at different times are assumed to be independent, the “values” of the resulting Markov process can be calculated. These values lead to information about the transient behavior, autocorrelation function, expected first passage time, and expected extra delay that arises if another unit is inserted into the system. An expression for the geometric transform or moment generating function of the probability distribution of the total service time of units in the system is determined. The results are derived for arbitrary arrival and service time distributions. In Part II, the continuous time analog of the system of Part I is considered and analogous results are determined.

Impact of Behavioral Factors on Performance of Multi-Server Queueing Systems

In this paper we consider a class of single-server queueing systems with compound Poisson arrivals, in which, at service completion epochs, the server has the option of taking off for one or several vacations of random length. The cost structure consists of a holding cost rate specified by a general non-decreasing function of the queue size, fixed costs for initiating and terminating service, and a variable operating cost incurred for each unit of time that the system is in operation. We show under some weak conditions with respect to the holding cost rate function and the service time, vacation time and arrival batch size distributions that it is either optimal among all feasible (stationary and non-stationary) policies never to take a vacation, or it is optimal to take a vacation when the system empties out and to resume work when, upon completion of a vacation, the queue size is equal to or in excess of a critical threshold. These optimality results are generalized for several variants of this model.

We analyze the so-called shortest queue first (SQF) queueing discipline whereby a unique server addresses queues in parallel by serving at any time, the queue with the smallest workload. Considering a stationary system composed of two parallel queues and assuming Poisson arrivals and general service time distributions, we first establish the functional equations satisfied by the Laplace transforms of the workloads in each queue. We further specialize these equations to the so-called “symmetric case,” with same arrival rates and identical exponential service time distributions at each queue; we then obtain a functional equation $$\begin{aligned} M(z) = q(z) \cdot M \circ h(z) + L(z) \end{aligned}$$ for unknown function M, where given functions $$q, L$$ , and $$h$$ are related to one branch of a cubic polynomial equation. We study the analyticity domain of function M and express it by a series expansion involving all iterates of function $$h$$ . This allows us to determine empty queue probabilities along with the tail of the workload distribution in each queue. This tail appears to be identical to that of the head-of-line preemptive priority system, which is the key feature desired for the SQF discipline.

Algorithm for computing the queue length distribution at various time epochs in DMAP/G(1, a, b)/1/N queue with batch-size-dependent service time

This work investigates a remote status updating system where the transmission process is modeled as a multi-stream M/G/1/1 non-preemptive system. We derive the closed-form expression of the average AoI of each stream in a heterogeneous case, where the distributions of service time are different for streams. To obtain more insights, we apply the results in a homogeneous system, where the service time distributions are identical, and find that preemption of packets would not always lead to the reduction of AoI, especially when the variance coefficient of the service time is small. We further optimize the generation rate to minimize the sum of average AoI. The results in heterogeneous cases show that given the same average service time for all streams, a higher generation rate should be allocated to the stream with a small service time variance. For the homogeneous cases with different AoI urgency weights for each stream, a higher generation rate should be reserved for the stream with more urgent AoI requirements for timeliness improvement. Besides, a lower bound on sum of average AoI is also provided, which only depends on the service rate and the number of data streams in homogeneous systems. Numerical results validate our theoretical analysis.

An analysis of the equilibrium strategies for route-choosing customers in a two-station queueing system

An analytic model of a single processor scheduling problem is investigated. The scheduling objective is to minimize the total loss incurred by a finite number of initially available requests when each request has an associated linear loss function. The assumptions of the model are that preemption is allowed with negligible loss of processor time, and that the distribution of actual service times is known for each class of requests. A request is associated with a class by any of its characteristics except its actual service time. A contrived example demonstrates that one reasonable scheduling rule does not always minimize expected total loss. The major results of the paper are the definition of a new scheduling rule based on the known service time distributions, and the proof that expected total loss is always minimized by using this new rule. Brief consideration is given to generalizations of the model in which new requests arrive randomly, and preemption requires a non-negligible amount of processor time.

We review the principles of computer performance modelling with queues, and validate the models using the empirical data. We focus on modelling a special configuration of two computers working in tandem: a mirroring system. Distributed system under investigation has implemented as a series of Java applets communicating using the functions of java net.* package. The mirroring aspect of the system is modelled analytically with a difference queue. The difference queue is defined for a network of two queues in parallel, as the queue consisting of elements present in one queue and absent from the other queue. The queueing models under investigation are M/M/1, and M/D/1 queues. In our study we have observed that the service times were not completely random on a typical server and could not be approximated well by exponential distribution. Our results have shown that the service was almost (i.e. with a very small variance) deterministic. We have also found that the d-queue model approximates well the behaviour of the mirroring system under low load conditions, and also provides an upper bound on the delays under high load conditions.

This paper presents a discrete time bulk service queue under the policy (a, c, d) with state dependent service rates. The inter-arrival times are assumed to be independent and geometrically distributed. The customers are served by a single server under the policy (a, c, d) with state dependent service rates. In this model the server begins service only when there are at least c units in the queue and he serves a maximum of d units in a batch. The server continue to serve even when the queue size is less than c but not less than a secondary limit a after a service completion epoch, but with a different service rate. If after a service completion epoch the queue size is less than a, the server becomes idle. The service times are also assumed to be independent and geometrically distributed. The steady state probabilities and some performance measures of the model are computed.

We focus on correlated queues where service and inter-arrival times are strongly correlated, and the input and output link speeds are the same. We present an analytical method to derive the LST (Laplace Stieltjes transform) of the (actual) waiting time distribution. To investigate the effect of correlation between service and inter-arrival times on the system performance, we consider a counterpart GI/G/1 queue where inter-arrival time and service time distributions are the same as in our correlated system, but they are independent. Some numerical examples are provided to show that such correlations have significant impact on the system performance.

Sensors composed of light barriers and switching mats are used to continuously measure service times at security and border checkpoints in an airport setting. The service times in combination with the number of service checkpoints operating in parallel determine the passenger flow. Checkpoints are a major bottleneck at airports and hence determine the capacity of the whole airport system. The results show that at the considered locations both light barriers and switching mats after initial calibration are able to provide continuous information of acceptable accuracy on the distribution of service times. In addition, the technology can be used to inexpensively measure the flow of passengers with reasonable accuracy compared with other pedestrian counting technologies. As such it provides valuable input to the airport operations management.

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.

Vehicle routing problems with stochastic travel and service times (VRPSTT) consist of designing transportation routes of minimal expected cost over a network where travel and service times are represented by random variables. Most of the existing approaches for VRPSTT are conceived to exploit the properties of the distributions assumed for the random variables. Therefore, these methods are tied to a given family of distributions and subject to strong modeling assumptions. We propose an alternative way to model travel and service times in VRPSTT without making many assumptions regarding such distributions. To illustrate our approach, we embed it into a state-of-the-art routing engine and use it to conduct experiments on instances with different travel and service time distributions.

A cellular communication system is divided into two main parts, core network, and radio access network. This research is concerned with the radio access network part which consists of multiple-cells, each served by a central located base station. Furthermore, the users in each cell are considered to be uniformly distributed inside the cell. In the downlink context, the users’ packets usually arrive at the base station via fiber optic and then are relayed to the users via radio waves of certain frequency/ies. The speeds of delivering users’ packets vary, depending on the users’ location. In this paper, the actual distribution of the service time over different users whose locations are uniformly distributed in a cell served by one base station is analytically found. Simulation results are presented to validate the derived model.