On Little’s Formula in Multiphase Queues

The structure of this work in the field of queuing theory consists of two stages. The first stage presents Little’s Law in Multiphase Systems (MSs). To obtain this result, the Strong Law of Large Numbers (SLLN)-type theorems for the most important MS probability characteristics (i.e., queue length of jobs and virtual waiting time of a job) are proven. The next stage of the work is to verify the result obtained in the first stage.

A M/M/1 Queueing Network with Classical Retrial Policy

In this paper we study the M/M/1 queueing model with retrial on network. We derive the steady state probability of customers in the network, the average number of customers in the all the three nodes in the system, the queue length, system length using little’s formula. The particular case is derived (no retrial). The numerical examples are given to test the correctness of the model.

Analysis of Single-Server Queue with Phase-Type Service and Energy Harvesting

We propose a queueing model suitable, for example, for modelling operation of nodes of sensor networks. The sensor node senses a random field and generates packets to be transmitted to the central node. The sensor node has a battery of a finite capacity and harvests energy during its operation from outside (using solar cells, wind turbines, piezoelectric cells, etc.). We assume that, generally speaking, service (transmission) of a packet consists of a random number of phases and implementation of each phase requires a unit of energy. If the battery becomes empty, transmission is failed. To reduce the probability of forced transmission termination, we suggest that the packet can be accepted for transmission only when the number of energy units is greater than or equal to some threshold. Under quite general assumptions about the pattern of the arrival processes of packets and energy, we compute the stationary distributions of the system states and the waiting time of a packet in the system and numerically analyze performance measures of the system as functions of the threshold. Validity of Little’s formula and its counterpart is verified.

Note on generalizations of Mecke's formula and extensions of H = λG

Mecke's formula is concerned with a stationary random measure and shift-invariant measure on a locally compact Abelian group , and relates integrations concerning them to each other. For , we generalize this to a pair of random measures which are jointly stationary. The resulting formula extends the so-called Swiss Army formula, which was recently obtained as a generalization for Little's formula. The generalized Mecke formula, which is called GMF, can be also viewed as a generalization of the stationary version of H = λG. Under the stationary and ergodic assumptions, we apply it to derive many sample path formulas which have been known as extensions of H = λG. This will make clear what kinds of probabilistic conditions are sufficient to get them. We also mention a further generalization of Mecke's formula.

Note on generalizations of Mecke's formula and extensions of H = λG

Mecke's formula is concerned with a stationary random measure and shift-invariant measure on a locally compact Abelian group, and relates integrations concerning them to each other. For, we generalize this to a pair of random measures which are jointly stationary. The resulting formula extends the so-called Swiss Army formula, which was recently obtained as a generalization for Little's formula. The generalized Mecke formula, which is called GMF, can be also viewed as a generalization of the stationary version ofH = λG. Under the stationary and ergodic assumptions, we apply it to derive many sample path formulas which have been known as extensions ofH = λG.This will make clear what kinds of probabilistic conditions are sufficient to get them. We also mention a further generalization of Mecke's formula.

Sample-path stability conditions for multiserver input-output processes

We extend our studies of sample-path stability to multiserver input-output processes with conditional output rates that may depend on the state of the system and other auxiliary processes. Our results include processes with countable as well as uncountable state spaces. We establish rate stability conditions for busy period durations as well as the input during busy periods. In addition, stability conditions for multiserver queues with possibly heterogeneous servers are given for the workload, attained service, and queue length processes. The stability conditions can be checked from parameters of primary processes, and thus can be verified a priori. Under the rate stability conditions, we provide stable versions of Little's formula for single server as well as multiserver queues. Our approach leads to extensions of previously known results. Since our results are valid pathwise, non-stationary as well as stationary processes are covered.