Complete characterisation of the customer delay in a queueing system with batch arrivals and batch service

Whereas the buffer content of batch-service queueing systems has been studied extensively, the customer delay has only occasionally been studied. The few papers concerning the customer delay share the common feature that only the moments are calculated explicitly. In addition, none of these surveys consider models including the combination of batch arrivals and a server operating under the full-batch service policy (the server waits to initiate service until he can serve at full capacity). In this paper, we aim for a complete characterisation-i.e., moments and tail probabilities - of the customer delay in a discrete-time queueing system with batch arrivals and a batch server adopting the full-batch service policy. In addition, we demonstrate that the distribution of the number of customer arrivals in an arbitrary slot has a significant impact on the moments and the tail probabilities of the customer delay

Delay analysis of a two-class batch-service queue with class-dependent variable server capacity

In this paper, we analyse the delay of a random customer in a two-class batch-service queueing model with variable server capacity, where all customers are accommodated in a common single-server first-come-first-served queue. The server can only process customers that belong to the same class, so that the size of a batch is determined by the length of a sequence of same-class customers. This type of batch server can be found in telecommunications systems and production environments. We first determine the steady state partial probability generating function of the queue occupancy at customer arrival epochs. Using a spectral decomposition technique, we obtain the steady state probability generating function of the delay of a random customer. We also show that the distribution of the delay of a random customer corresponds to a phase-type distribution. Finally, some numerical examples are given that provide further insight in the impact of asymmetry and variance in the arrival process on the number of customers in the system and the delay of a random customer

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

Tail distribution of the delay in a general batch-service queueing model

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Discrete Time Analysis of Multi-Server Queueing Systems in Material Handling and Service

In this doctoral thesis, performance parameters of multi-server queueing systems are estimated under general stochastic assumptions. We present an exact calculation method for the discrete time distribution of the number of customers in the queueing system at the arrival moment of an arbitrary customer. The waiting time distribution and the sojourn time distribution are estimated exactly, as well. For the calculation of the inter departure time distribution, we present an approximation method

Order batching in multi-server pick-and-sort warehouses.

In many warehouses, customer orders are batched to profit from a reduction in the order picking effort. This reduction has to be offset against an increase in sorting effort. This paper studies the impact of the order batching policy on average customer order throughput time, in warehouses where the picking and sorting functions are executed separately by either a single operator or multiple parallel operators. We present a throughput time estimation model based on Whitt's queuing network approach, assuming that the number of order lines per customer order follows a discrete probability distribution and that the warehouse uses a random storage strategy. We show that the model is adequate in approximating the optimal pick batch size, minimizing average customer order throughput time. Next, we use the model to explore the different factors influencing optimal batch size, the optimal allocation of workers to picking and sorting, and the impact of different order picking strategies such as sort-while-pick (SWP) versus pick-and-sort (PAS)Order batching; Order picking and sorting; Queueing; Warehousing;

System occupancy of a two-class batch-service queue with class-dependent variable server capacity

Due to their wide area of applications, queueing models with batch service, where the server can process several customers simultaneously, have been studied frequently. An important characteristic of such batch-service systems is the size of a batch, that is the number of customers that are processed simultaneously. In this paper, we analyse a two-class batch-service queueing model with variable server capacity, where all customers are accommodated in a common first-come-first served single-server queue. The server can only process customers that belong to the same class, so that the size of a batch is determined by the number of consecutive same-class customers. After establishing the system equations that govern the system behaviour, we deduce an expression for the steady-state probability generating function of the system occupancy at random slot boundaries. Also, some numerical examples are given that provide further insight in the impact of the different parameters on the system performance

Discrete Time Analysis of Consolidated Transport Processes

Diese Arbeit beschäftigt sich mit der Entwicklung zeitdiskreter Modelle zur Analyse von Transportbündelungen. Mit den entwickelten Modellen für Bestands- und Fahrzeugbündelungen, insbesondere Milkrun-Systeme, kann eine detaillierte Leistungsbewertung in kurzer Zeit durchgeführt werden. Darüber hinaus erlauben die Modelle die Analyse der Umschlagslagerbündelungen, beispielweise Hub-und-Spoke-Netzwerke, indem sie im Rahmen einer Netzwerkanalyse mit einander verknüpft werden