Overlap Times in the GIB/GI/∞GI^B/GI/\infty Queue

Overlap times have been studied as a way of understanding the time of interaction between customers in a service facility. Most of the previous analysis relies on the single jump assumption for arrivals, which implies the queue increases by one for each arrival epoch. In this paper, we relax the single arrival assumption and explore the impact of having batch arrivals. Unfortunately, with batch arrivals it is not clear how one measures an overlap time between batches of customers. Thus, we develop two ways of capturing the notion of an overlap time in a batch setting and derive exact results in the infinite server queue with batch arrivals. Finally, we derive new results for analyzing overlap times of more than two batches

Non-Stationary Queues with Batch Arrivals

Motivated by applications that involve setting proper staffing levels for multi-server queueing systems with batch arrivals, we present a thorough study of the queue-length process $\{Q(t); t \geq 0\}$, departure process $\{D(t); t \geq 0\}$, and the workload process $\{W(t); t \geq 0\}$ associated with the M$_{t}^{B_{t}}$/G$_{t}$/$\infty$ queueing system, where arrivals occur in batches, with the batch size distribution varying with time. Notably, we first show that both $Q(t)$ and $D(t)$ are equal in distribution to an infinite sum of independent, scaled Poisson random variables. When the batch size distribution has finite support, this sum becomes finite as well. We then derive the finite-dimensional distributions of both the queue-length process and the departure process, and we use these results to show that these finite-dimensional distributions converge weakly under a certain scaling regime, where the finite-dimensional distributions of the queue-length process converge weakly to a shot-noise process driven by a non-homogeneous Poisson process. Next, we derive an expression for the joint Laplace-Stieltjes transform of $W(t)$, $Q(t)$, and $D(t)$, and we show that these three random variables, under the same scaling regime, converge weakly, where the limit associated with the workload process corresponds to another Poisson-driven shot-noise process

Analysis of the transient delay in a discrete-time buffer with batch arrivals

We perform a discrete-time analysis of the delay of customers in a FIFO buffer with batch arrivals. The numbers of arrivals per slot are independent and identically distributed variables. Since the arrivals come in batches, the delays of the subsequent customers do not constitute a Markov chain, which complicates the analysis. By using generating functions and the supplementary variable technique, moments of the delay of the k-th customer are calculated

Delay analysis of two batch-service queueing models with batch arrivals: Geo(X)/Geo(c)/1

In this paper, we compute the probability generating functions (PGF's) of the customer delay for two batch-service queueing models with batch arrivals. In the first model, the available server starts a new service whenever the system is not empty (without waiting to fill the capacity), while the server waits until he can serve at full capacity in the second model. Moments can then be obtained from these PGF's, through which we study and compare both systems. We pay special attention to the influence of the distribution of the arrival batch sizes. The main observation is that the difference between the two policies depends highly on this distribution. Another conclusion is that the results are considerably different as compared to Bernoulli (single) arrivals, which are frequently considered in the literature. This demonstrates the necessity of modeling the arrivals as batches

Beyond Model-Checking CSL for QBDs: Resets, Batches and Rewards

We propose and discuss a number of extensions to quasi-birth-death models (QBDs) for which CSL model checking is still possible, thus extending our recent work on CSL model checking of QBDs. We then equip the QBDs with rewards, and discuss algorithms and open research issues for model checking CSRL for QBDs with rewards