Block-structured supermarket models

Abstract

Supermarket models are a class of parallel queueing networks with an adaptive control scheme that play a key role in the study of resource management of, such as, computer networks, manufacturing systems and transportation networks. When the arrival processes are non-Poisson and the service times are non-exponential, analysis of such a supermarket model is always limited, interesting, and challenging. This paper describes a supermarket model with non-Poisson inputs: Markovian Arrival Processes (MAPs) and with non-exponential service times: Phase-type (PH) distributions, and provides a generalized matrix-analytic method which is first combined with the operator semigroup and the mean-field limit. When discussing such a more general supermarket model, this paper makes some new results and advances as follows: (1) Providing a detailed probability analysis for setting up an infinitedimensional system of differential vector equations satisfied by the expected fraction vector, where the invariance of environment factors is given as an important result.(2) Introducing the phase-type structure to the operator semigroup and to the meanfield limit, and a Lipschitz condition can be obtained by means of a unified matrixdifferential algorithm. (3) The matrix-analytic method is used to compute the fixed point which leads to performance computation of this system. Finally, we use some * The main results of this paper will be published in "Discrete Event Dynamic Systems" 2014. On the other hand, the three appendices are the online supplementary material for this paper published in "Discrete Event Dynamic Systems" 2014 1 numerical examples to illustrate how the performance measures of this supermarket model depend on the non-Poisson inputs and on the non-exponential service times.Thus the results of this paper give new highlight on understanding influence of nonPoisson inputs and of non-exponential service times on performance measures of more general supermarket models.

Methodology of probabilistic analysis of state dynamics of multi­dimensional semi­Markov dynamic systems

“…The model contains a set of integral equations for the unknown functions describing the probabilistic dynamics of the system (1) - (4). The solution of these integral equations is obtained using the Laplace transform (5)- (13). As a result of solving the integral equations, the desired relation (41) is obtained for calculating the conditional probability of the object stay in the state 0 H at an arbitrary time t if the object was in the state 0 H at the initial time.…”

“…The decision regarding the efficiency of the system is made on the basis of the final state probability distribution obtained. [13] explores a queuing system with non-Poisson arrivals and non-exponential service in order to obtain stationary performance characteristics. In [14], a queuing system with semi-Markov arrivals is investigated.…”

Data-Driven Simulation Modeling of the Checkout Process in Supermarkets: Insights for Decision Support in Retail Operations

“…However, due to data limitations or for the sake of simplicity some authors do not distinguish particular steps and treat service as a single step that covers all three actions. Regarding simulations, supermarket models typically assume exponential service times [34], but other distributions -like triangular [35], lognormal [36] or phase-type (PH) [37] have been considered as well. If we want to be more precise, we can decompose the service time into two factors that drive it: work amount and cashier speed.…”

An Overview for Markov Decision Processes in Queues and Networks

“…Load balancing is an interesting research direction in queueing networks with simply observable information, e.g., see Down and Lewis [103], Chou and Abraham [79], e Silva and Gerla [288], Li and Kameda [217], Li et al [220,221], Li [219] and Li and Lui [224].…”