Energy-Aware M^X/G/1 Queue

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

In queueing theory, an //1 queue (in Kendall notation) is a model in which arrivals satisfy a Markovian distribution , service times have an unknown distribution, called a general distribution , and there is a single server. Additional notation: PS indicates that there is processor sharing on the server, while indicates that the arrival distribution is batch Markovian with a random number of simultaneous arrivals.


//1-PS queueing systems are well-known models in the performance analysis of various systems, ranging from telecommunication channels to air transportation systems. Consider an energy-aware //1-PS system, which may possibly be put in a sleep state to save energy whenever it is idle. In most cases, such a system needs some setup time before it "wakes up" from a sleep state to process requests. This Demonstration illustrates how the number of requests of such a system evolve through time. Statistics for mean values are provided in the lower panel.


Contributed by: Misikir Eyob Gebrehiwot (October 2015)
Open content licensed under CC BY-NC-SA



Snapshot 1: no setup delay (cost) incurred to turn system on from sleep, and system is allowed to remain idle

Snapshot 2: no setup delay (cost), and system is put to sleep immediately when it becomes idle (idle time is 0)

Snapshot 3: setup delay is 5 time units, and system is put to sleep whenever it is idle for 10 time units (idle time is 10)

In Snapshot 1, the setup delay from sleep to busy state is zero (server can be set up immediately). But the system does not benefit from this situation because it does not go to sleep when it is idle, thus resulting in a wasted opportunity to save energy.

In Snapshot 2, the setup delay is zero, and the system is switched to sleep state immediately when it becomes idle, giving a much lower mean power consumption compared to the scenario in Snapshot 1. Note that mean response time roughly remains the same as in Snapshot 1, any variation coming only from the stochastic nature of the simulation.

In Snapshot 3, it takes 5 time units to start the server from sleep, and the system has an idling timer that puts the server to sleep when it expires. This gives longer response times and slightly lower power consumption.

More on //1 queues can be found in [1], and more on energy-performance tradeoff in //1 systems in [2].


[1] L. Kleinrock, Queueing Systems, Volume I: Theory, New York: Wiley, 1975.

[2] M. Gebrehiwot, S. Aalto, and P. Lassila, "Optimal Sleep-State Control of Energy-Aware M/G/1 Queues," in Proceedings of the 8th International Conference on Performance Evaluation Methodologies and Tools, Brussels: ICST, 2014 pp. 82–89.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.