Corrections to the article ‘On the analytical model for jitter’ by Nadarajah and Ba

Introduction

Given the importance of jitter to the quality of service in real-time applications and the lack of simple analytical models that accurately capture this performance metric, we have developed several analytical models since 2009 to describe jitter behavior under various scenarios. These models range from simple queuing systems, such as M/M/1 with Poisson traffic (Dahmouni et al., 2012), to more complex systems, such as G/M/1 queues (Dbira et al., 2016), which account for non-Poisson traffic.

Jitter is a measure of the variation of a packet’s transfer delay and is defined by the IETF (Demichelis and Chimento, 2002) as follows. Let T i represent the delay experienced by packet i going through a queue. The average jitter is then given by the expected value of the absolute jitter:

J = E T i + 1 − T i . (1)

In Section 3 of our paper, Dahmouni et al. (2012), we developed analytical formulas for the jitter of a tagged traffic stream, e.g., all the packets of a particular movie being watched by a user, as a function of its load. The authors of Nadarajah and Ba (2025) claim that our formulas (9–12) are incorrect and propose an alternative expression, which they present as more general. They make the following claims.

• Claim 1: There is some confusion in Dahmouni et al. (2012) regarding the queue being investigated.
• Claim 2: Our formula J = 1 μ, developed in Dahmouni et al. (2012), is valid only when η = μ − λ. They propose Equation 2 as a general formulation.
• Claim 3: Their expression (Equation 2) is a more general formula for the jitter as it can be applied to queues other than M/M/1.

We now explain why these claims appear inconsistent with established results.

On claim 1: confusion regarding our gueue model

In Dahmouni et al. (2012), we specifically state that we consider Poisson traffic and exponential service times in the abstract, in Sections 1.1 and 1.3, and at the beginning of Section 3. In queuing theory, this defines an M/M/1 queue using Kendall’s notation (Kleinrock, 1975). Thus, the queueing model is specified as an M/M/1 queue under the stated assumptions of Poisson arrivals and exponential service times.

On claim 2: the role of η

Their proposed extended jitter formula

The authors of Nadarajah and Ba (2025) propose the following Formula 2 for computing the jitter

J = η + λ 2 + μ 2 η + λ μ η + μ + λ . (2)

This is derived from Equation 7 of Dahmouni et al. (2012) under the following assumptions:

• the arrival process is exponential with parameter λ;
• the service time is exponential with parameter μ; and
• the transit time is exponential with parameter η;

In Equation 2, η is viewed as a parameter independent of λ and μ. The authors further indicate that their formula is more general than ours since our result J = 1 μ is a special case of Equation 2, which can be derived as follows. Setting η = μ − λ, the two terms in the denominator becomeSubstituting them in Equation 2, we obtain

η + λ = μ η + λ + μ = 2 μ .

J = η + λ 2 + μ 2 η + λ μ η + μ + λ = μ 2 + μ 2 μ ⋅ μ ⋅ 2 μ = 2 μ 2 2 μ 3 = 1 μ . (3)

Our response:

First, recall the fundamental assumptions of an M/M/1 queue and the relationship between η, λ, and μ, as can be found in Kleinrock (1975) and Stewart (2009).

An M/M/1 queue is defined by exponential inter-arrival times I and also exponential service times S. where λ is the arrival rate and μ is the service rate, and the usual stability condition λ < μ holds.

I ∼ E x p λ , with f I t = λ e − λ t , t ≥ 0 ,

S ∼ E x p μ , with f S t = μ e − μ t , t ≥ 0 ,

Under these assumptions, it is well known (Kleinrock, 1975; Stewart, 2009) that the system (or transit) time T follows an exponential distribution with parameter η = μ − λ and that the mean transit time is given by

E T = 1 μ − λ = 1 η . (4)

In other words, for the M/M/1 queue, the parameter η is fully determined by the traffic parameters λ and μ and, therefore, cannot be considered independent: the only admissible value is η = μ − λ. Substituting Equation 4 into Equation 2 yields J = 1 μ, which is our result.

On claim 3: generalization to other queues

Even though the derivation of Equation 2 relies on the M/M/1 assumption, the authors claim that it can be applied to other types of queues, for instance, when either the arrival or the service process is not exponential. This has been studied in Dbira et al. (2016), and we take the simple case of a general arrival and exponential service, a G/M/1 queue, to show how the jitter can be computed for non-Poisson queues.

We know from Kleinrock (1975) that the transit time T of a G/M/1 queue follows an exponential distribution where η is the transit rate and τ is the probability of waiting, given by the unique root in the interval (0, 1) ofwhere F I ( s ), the Laplace transform of the inter-arrival distribution I ( y ), is given by

f T x = η e − η x , (5)

η = μ 1 − τ , (6)

τ = F I μ − μ τ , (7)

F I s = ∫ 0 ∞ f I y e − s y d y . (8)

A very important point is that η is not an independent parameter but instead depends entirely on the parameters of the inter-arrival and service-time distributions. From this, we can show thatwhich is very different from Equation 2.

J = η 2 + μ 2 η μ η + μ + 2 η + μ F I η + μ − 1 η F I η , (9)

Consider now the case where the inter-arrival distribution is a Gamma, G m ( x ; k , θ ), with shape k > 0 and scale θ > 0. We haveandwhich can be replaced in Equation 9. This means that J ( k , θ , μ ) is a function of three independent parameters: μ for the service time and k and θ for the arrival times. In that expression, η ( k , θ , μ ) is also a function of the same parameters from Equations 4–8, 10, 11.

f I x = x k − 1 e − x / θ θ k Γ k , (10)

F I s = 1 1 + s θ k , (11)

In Nadarajah and Ba (2025), on the other hand, Equation 2 is presented as a function of three independent parameters μ, λ, and η. We know, however, that η is not an independent variable, which leaves an equation with two independent variables λ and μ, while the queue is characterized by three independent variables k, θ, and μ.

For this reason, it is not clear at all how Equation 2 could be used to compute the jitter of a G/M/1 queue. This holds for any queue in which the total number of parameters describing the arrival and service processes is greater than 2.

Conclusion

To summarize,

• We have clearly stated that our analysis is based on exponential inter-arrival and service times, i.e., the M/M/1 queue model.
• Within this framework, the expression J = 1 μ, follows directly and is consistent with the definition of jitter under these assumptions.
• The interpretation of expression (Equation 2) as a generalization for any η appears to rely on treating η as an independent parameter. However, in the M/M/1 setting used to derive (Equation 2), η is a function of λ and μ, which limits the generalization of (Equation 2) to other η values.
• The derivation of (Equation 2) relies explicitly on the exponential form of the arrival and service processes. Further justification would be required before applying this expression to non-exponential arrival or service processes.
• The example of the G/M/1 shows that expression (Equation 2) differs from the known results Dbira et al. (2016), which further suggests that its use beyond the M/M/1 case may be limited.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

Edited by: Koffka Khan, The University of the West Indies St. Augustine, Trinidad and Tobago

Reviewed by: Santosh I. Gore, Sai Info Solution, India