There are many applications where it is necessary to model queuing systems that involve finite queue size. Most of the models consider traffic with Poisson arrivals and exponentially distributed service times. Unfortunately, when the traffic behavior does not consider Poisson arrivals and exponentially distributed service times, closed-form solutions are not always available or have high mathematical complexity. Based on Lindley’s recursion, this paper presents a fast simulation model for an accurate estimation of the performance metrics of G/G/1/K queues. One of the main characteristics of this approach is the support for long-range dependence traffic models. The model can be used to model queuing systems in the same way that a discrete event simulator would do it. This model has a speedup of at least two orders of magnitude concerning implementations in conventional discrete event simulators.
Traditionally, queuing theory applications are limited to systems with assumptions that derive in closed-form expressions. For example, most of them restrict the tractability of solutions or the tools available to achieve numerical results. Some features that impact tractability are queue size, arrival process, service time distribution, and queue service’s policy. In terms of queue size, the queuing systems solutions divided into infinite and finite queue size.
For practical reasons, real-world applications in manufacturing, transportation, communication, networking, and computation systems have finite queues [
In the same way, arrival processes can be loosely classified by the decay of its autocorrelation function in ones with Short Range Dependence (SRD), i.e., fast decay, and ones with Long-Range Dependence (LRD) or slow decay [
Service time distribution broadly classifies queuing systems in exponentially distributed (Markovian property, i.e., memoryless,
On the other hand, simulations can solve queuing systems with no closed expressions but, unfortunately, have high computation times in many cases. A fast discrete event simulation model for a priority round robin multiplexer based on Lindley-type recursions is presented in [
The rest of the paper is organized as follows. Section
This section introduces a brief description of some analytical results for finite queuing systems.
As mentioned in the previous section, there are few results for the
The blocking probability
The average response time that is the expected sojourn time of an entity (in other words, the expected waiting time of an entity given that the entity must wait in the queue) can be derived from Little’s theorem and using equations (
The method is based on the consideration that the M/G/1/K queueing system can be seen as an imbedded-Markov-chain observing the number of entities left behind upon the departure of an entity [
Distribution of the number of arrivals during a service time
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Exponential | Erlang-2 | Deterministic |
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Modern investigations of traffic measurements suggest that self-similar processes and Long-Range dependence (LRD) can be applied to the study and accurate modeling of network traffic [
Let
Figure
Aggregated processes,
Then it is said that
The variance of the aggregated time series is defined by equation (
The plot
This approach is based on difference equations based on Lindley’s recursion, in order to model the waiting and service times of a queuing system. Performance metrics such as mean service time and mean queue size can be calculated numerically from the data processed by the difference equations. This process is the same as any simulation software executes. However, this approach takes off all the unnecessary details needed for a full-scale simulation. In the Fast Discrete Event Simulation (FDES) approach we call every customer an entity because it can be a client, packet, call, etc. depending on the system modeled. However, FDES only takes into consideration the arrival and service times for every entity that comes into the system. Then FDES takes these two quantities and processes them into the recursive equations to obtain the waiting and departure times for every entity.
The model consists of a
Model of a queueing system.
Let
Figure
Time diagram for the queueing system and the queue size.
We define the random variable
From the bottom of Figure
The description of Algorithm
In the previous section, we presented a
The model consists of a
In order to estimate the performance metrics of the
The blocking probability
This section presents numerical results for the performance analysis of the
Exact (or theoretical) results are obtained as follows (unless otherwise stated): the
Table
Average results for G/G/1/K queueing systems,
Results (Average) | | | | | Runtime (sec.) |
---|---|---|---|---|---|
| 1.0246 | 0.4468 | 0.4223 | 0.0371 | 2.4538e-04 |
| 1.0263 | 0.4473 | 0.4245 | 0.0363 | 9.4462 |
| 1.0263 | 0.4473 | 0.4245 | 0.0363 | 0.0193 |
| |||||
| 0.8544 | 0.2574 | 0.6844 | 0.0051 | 2.3045e-04 |
| 0.8547 | 0.2585 | 0.6802 | 0.0050 | 9.3824 |
| 0.8547 | 0.2585 | 0.6802 | 0.0050 | 0.0190 |
t: theoretical; s: simulation; f: FDES.
Likewise, Table
Average results for G/G/1/K queueing systems,
Results (Average) | | | | | | |
---|---|---|---|---|---|---|
| 1 | 0.5 | 1.0784 | 0.9044 | 0.4337 | 0.0562 |
| 1 | 0.5146 | 1.0813 | 0.9091 | 0.4306 | 0.0571 |
| 1 | 0.5146 | 1.0813 | 0.9091 | 0.4306 | 0.0571 |
| ||||||
| 1 | 0.5109 | 1.0806 | 0.9106 | 0.4311 | 0.0577 |
| 1 | 0.5109 | 1.0806 | 0.9106 | 0.4311 | 0.0577 |
| ||||||
| 5 | 0.6259 | 3.4982 | 2.5002 | 0.0117 | 0.6629 |
| 5 | 0.6259 | 3.4982 | 2.5002 | 0.0117 | 0.6629 |
| ||||||
| 10 | 0.7618 | 3.7941 | 2.7763 | 0.0018 | 0.8321 |
| 10 | 0.7618 | 3.7941 | 2.7763 | 0.0018 | 0.8321 |
t: theoretical; s: simulation; f: FDES.
Figures
Average waiting time in the queue for the
Blocking probability for different values of
Table
Average results for G/G/1/K queueing systems,
Results (Average) | | | | | | |
---|---|---|---|---|---|---|
| 1.2 | 0.9 | 1.0719 | 0.8962 | 0.4340 | 0.0554 |
| 1.2 | 0.9 | 1.0719 | 0.8962 | 0.4340 | 0.0554 |
| ||||||
| 1.8 | 0.6 | 1.0698 | 0.8884 | 0.4386 | 0.0542 |
| 1.8 | 0.6 | 1.0698 | 0.8884 | 0.4386 | 0.0542 |
| ||||||
| 1.2 | 0.9 | 1.1875 | 1.6893 | 0.1525 | 0.2646 |
| 1.2 | 0.9 | 1.1875 | 1.6893 | 0.1525 | 0.2646 |
| ||||||
| 1.8 | 0.6 | 1.0631 | 0.8385 | 0.4715 | 0.0363 |
| 1.8 | 0.6 | 1.0631 | 0.8385 | 0.4715 | 0.0363 |
s: simulation; f: FDES.
Figures
Blocking probability for different values of
Average waiting time in the queue for different values of
Other interesting results can be observed when analyzing a type of process similar to the PPBP model. Consider that, in PPBP, burst arrivals occur as Poisson processes with Pareto distributed length. In PPBP a burst is divided into small pieces at a rate
Figures
Blocking probability for different values of
Average waiting time in the queue for different values of
Finally, the average runtime for the different models was estimated and was shown in Figure
Runtime comparison for different implementations of the
On the other hand, Figure
Runtime comparison for different values of
Based on Lindley’s recursion, a fast simulation model for the performance analysis of the
No data were used to support this study.
The authors declare that they have no conflicts of interest.