The effect of the dependency in the Markov renewal arrival process on the various performance measures of an exponential server queue

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Virginia Polytechnic Institute and State University


The thesis of this paper is to investigate how the dependency in the arrival process affects the queueing performance measures. The Markov renewal arrival process (MRAP) was chosen as the arrival process. This choice was made because many of the typical arrival processes can be obtained as special cases of the MRAP. But the main reason behind this choice is that the interarrival times of the MRAP are dependent. We assume that the queue is a single server queue with exponential service time and the investigation was carried out numerically because no analytical solution was available.

There are 5 parameters of the arrival process used in this investigation: the traffic intensity (ρ), the squared coefficient of variation (scν), the serial correlation defined by the lag-1 correlation (corr) plus the rate ξ and the coefficient of skewness (𝛾). Here are the performance measures of the MR/M/1 queue we investigate: the expected queue length at arbitrary times (L𝓽), the standard deviation (σ) of the queue length at arbitrary times and the caudal characteristic η. The other performance measures such as: the expected queue length at arrival time, the waiting time, the sojourn time, etc. can be easily obtained from L𝓽. We compare these performance measures against those of the corresponding GI/M/1 queue.

When the lag-1 correlation of the arrival process is negative (this means that the lags of the serial correlation alternate in signs), the Lt of the MR/M/1 queue is smaller (but not by much) than the L𝓽 of the GI/M/1 queue. Therefore, we focus our attention to the MR/M/1 queue with positive serial correlation. The results are presented using graphs.

We find that the coefficient of skewness of the arrival process (𝛾) plays an important role. The L𝓽 curve decreases rapidly as 𝛾 increases and after certain values of 𝛾 called the turning region, the L𝓽 curves Hatten. This important observation indicates that to the left of the turning region, the L𝓽 is almost insensitive to the dependency in the arrival process. However, to the right of the turning region, the L𝓽 is sensitive to the positive serial correlation in the arrival process. Highly correlated arrival process (large corr and ξ) can cause the L𝓽 to be significantly larger than the L𝓽 for the uncorrelated queue.

For the MR/M/1 queue, the magnitude of the standard deviation σ is larger than the corresponding L𝓽. However, the shapes of the σ curves are similar to those of the L𝓽 curves. So, all of the conclusions drawn for the L𝓽 also apply to the standard deviation σ.

For the M/M/1 queue, the caudal characteristic η equals to the traffic intensity ρ (η=ρ). For the uncorrelated Gl/M/1 queue, one would expect that when scν<1.0, η<ρ (i.e., the queue would behave like a H/M/1 queue) and when scν>1.0, η>ρ (i.e., the queue would behave like a H/M/1 queue). Our results indicates that this is not necessarily true. We found again that the coefficient of skewness (𝛾) plays an important role. For the uncorrelated GI/M/1 queue with scν>1.0, η can be smaller than ρ when 𝛾 is large enough. For the correlated MR/M/1 queue, even for scν<1.0, a low 𝛾 value combined with the positive serial correlation can cause η to be larger than ρ. On the other hand, scν>1.0 does not necessarily results in η>ρ. A large value of 𝛾 can cause η to be smaller than ρ, even for the queue with highly correlated interarrival times.