Little’s Formula and Extensions

1999 ◽  
pp. 159-212
Author(s):  
Muhammad El-Taha ◽  
Shaler Stidham
Keyword(s):  
Little's Formula

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

2021 ◽  
Vol 12 (1S) ◽  
pp. 329-337
Author(s):  
S. Shanmugasundaram, Et. al.
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Queueing Network ◽  
State Probability ◽  
Queueing Model ◽  
Steady State Probability ◽  
Numerical Examples ◽  
System Length ◽  
Number Of Customers ◽  
Little's Formula

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.

Little’s Formula and Parking Behaviors

2018 ◽  
pp. 145-162
Author(s):  
Saburo Saito ◽  
Kosuke Yamashiro ◽  
Masakuni Iwami ◽  
Mamoru Imanishi
Keyword(s):  
Little's Formula

Time and customer processes in queues with stationary inputs

1977 ◽  
Vol 14 (02) ◽  
pp. 349-357 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Sufficient Conditions ◽  
Limiting Distributions ◽  
Queueing Processes ◽  
Little's Formula ◽  
General Conditions

Two types of processes occurring in queues with stationary inputs are considered. They are called ‘time processes’ and ‘customer processes’. Sufficient conditions for the convergence of sample averages and the existence of limiting distributions for each type of processes are given. The results generalize those of Loynes (1962). The results are applied to three queueing processes and the Little's formula L = λW is obtained under rather general conditions.

A Survey of J. Little's Formula

1983 ◽  
Vol 51 (3) ◽  
pp. 255 ◽  
Author(s):  
M. F. Ramalhoto ◽  
J. A. Amaral ◽  
M. Teresa Cochito
Keyword(s):  
Little's Formula

Simple derivations of the invariance relations and their applications

1982 ◽  
Vol 19 (1) ◽  
pp. 183-194 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
General Formula ◽  
Waiting Time ◽  
Simple Proof ◽  
Inversion Formula ◽  
Point Processes ◽  
Queueing Models ◽  
Unified Approach ◽  
Little's Formula

In the literature, various methods have been studied for obtaining invariance relations, for example, L = λW (Little's formula), in queueing models. Recently, it has become known that the theory of point processes provides a unified approach to them (cf. Franken (1976), König et al. (1978), Miyazawa (1979)). This paper is also based on that theory, and we derive a general formula from the inversion formula of point processes. It is shown that this leads to a simple proof for invariance relations in G/G/c queues. Using these results, we discuss a condition for the distribution of the waiting time vector of a G/G/c queue to be identical with that of an M/G/c queue.

scholarly journals Sample-path stability conditions for multiserver input-output processes

1994 ◽  
Vol 7 (3) ◽  
pp. 437-456 ◽  
Author(s):  
Muhammad El-Taha ◽  
Shaler Stidham
Keyword(s):  
A Priori ◽  
Sample Path ◽  
Stability Conditions ◽  
Single Server ◽  
Input Output ◽  
State Spaces ◽  
Multiserver Queues ◽  
Path Stability ◽  
The Stability ◽  
Little's Formula

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.

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

2016 ◽  
Vol 2016 ◽  
pp. 1-16 ◽  
Author(s):  
Sergey A. Dudin ◽  
Moon Ho Lee
Keyword(s):  
Sensor Node ◽  
Random Number ◽  
Single Server ◽  
Finite Capacity ◽  
Central Node ◽  
Phase Type ◽  
Server Queue ◽  
Arrival Processes ◽  
System States ◽  
Little's Formula

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.

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