8.2 Single-type queueing networks
8.2.2 Numerical results with a tandem network
For the first numerical study let us consider a simple2-node tandem queueing network as shown by Figure36. Traffic entering the network is directed to node1, after getting served it is forwarded to node2.
The parameters of the system are taken from real traffic measurements. For the first sce-nario the inputMAPhas been created from the inter-arrival times of the LBL trace based on 5marginal and2×2joint moments using the moment matching procedure described in Sec-tion3.2. The matrices of thisMAPare
D0(LBL)= With these parameters the arrival rate of the packets isλ = 188.29, the squared coefficient of variation of the inter-arrival times isc2A=2.183, and the lag-1auto-correlation is0.159.
The same LBL trace file contains information on the packet sizes as well. Matching the first two moments of the packet sizes we got the followingPHdistribution:
σ=h0.19937 0.80063
by which the mean packet size is138.86and the squared coefficient of variation is2.508. The packet size distribution and the speed of the transmission lineCiat nodeitogether determine the service process of the packets as
S0(i) =Ci·S,
S1(i) =Ci·(−S)1·σ.
The values ofCiare set such that utilization of both queues are equal to the desired valueρ.
All the methods for departure process approximation introduced in Section8.2.1are com-pared with simulation results in Figure37, which depicts the mean queue length at node2 as the function of the utilization. In the figure, the ”Joint moment based” curve covers all methods based on the lag-1joint moments, they gave exactly the same results in this partic-ular example. The lag-1joint moment based procedure has been tested both with matching 4(marginal- and joint-) moments and with matching9moments, the difference between the results were marginal in this queueing network. As expected, the Poisson approximation per-formed worst, and the lag-1based methods turned out to be the most accurate. The simple approximation based on the scaling of the service process was surprisingly accurate in this test case, but looking at the low utilization cases (right plot in Figure37) reveals the superior-ity of the lag-1based methods.
In Figure 38, comparing the squared coefficient of variation (SCV) of the number of cus-tomers, the lag-1joint moment based method achieved the highest accuracy again. In this
8.2 single-type qeueing networks 133 case, however, increasing the number of joint moments to match had a positive impact on the accuracy, the one matching9moments managed to reproduce the simulation results almost perfectly.
Table7provides further interesting properties of the algorithms. The second column con-tains the size of theMAPmodeling the departure process. It does not depend on the size of the arrival and service processes in the Poisson and in the lag-1joint moment based methods.
The ETAQA truncation method produces the largest departure process, at the minimal trun-cation level (at level2) it needs 18 phases, however, to improve accuracy, the truncation level should be increased. The number of non-Markovian results from the 36 executions (vary-ing the utilization between0.2and0.9) is given in the second column. The ETAQA method did not manage to produce a Markovian representation in any of the cases. The lag-1based method matching9moments occasionally returned a non-Markovian result as well, which were possible to fix with the algorithms described in Section3.3. The last column indicates the mean relative absolute error compared to the simulation results for the mean number of customers and for the squared coefficient of variation (SCV) of the number of customers at node2(separated by a slash character). The most accurate results are given by the ETAQA method with a high truncation level, but it is a huge and non-Markovian representation which is not tractable analytically.
In this scenario the input was taken from real traffic measurements, the packet inter-arrival times were relatively busty (with squared coefficient of variation2.183) and positively cor-related. In the next scenario the input will be a synthetically generated MAPthat has the
“opposite” behavior: it is more deterministic (the squared coefficient of variation is0.7278), and negatively correlated (the lag-1auto-correlation is−0.2119). The corresponding matrix parameters are
The packet size distribution is the same as before.
The mean and the squared coefficient of variation of the number of customers at node2is de-picted in Figures39and40. (The Poisson and the scaled service process based approximations are omitted to make it easier to distinguish between the curves). In this scenario the Lag-1 based method results in a non-valid stochastic process in many cases. The performance mea-sures returned by the queueing analysis are invalid if the queue is fed by an invalid stochas-tic process, which is clearly visible in the figure. This is a clear example demonstrating how important it is to stick with Markovian representations, and how essential the role of Algo-rithms4and5is. These algorithms, by fitting an invalid process with a valid one, managed to produce aMAPwith only very slightly different statistics, that approximate the queue length behavior of node2with a reasonable accuracy.
As visible in Table 8, this high accuracy is achieved by a compact representation, with 2 or 3 states only. Since the moment matching returned invalid processes, the difference between Algorithms4and5becomes visible, they follow different approaches to approximate the invalid process. In this particular example the accurate fitting of the joint moments turned out to be a slight better strategy, but both perform similarly well.
134 qeueing network analysis based on the joint moments
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 10 20 30
Utilization of both nodes
Meannumberofcustomers
Simulation Lag-1based Busy period based Poisson approximation
Scaled service process
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0
1 2 3
Utilization of both nodes
Meannumberofcustomers
Simulation Lag-1based Busy period based Poisson approximation
Scaled service process
Figure 37.: Mean number of customers at node2of the tandem network with LBL input
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2 4 6 8
Utilization of both nodes
SCVofthenumberofcustomers
Simulation Lag-1based Busy period based Poisson approximation
Scaled service process
Figure 38.: Squared coefficient of variation of the number of customers at node2of the tandem network with LBL input
Approximation algorithm # of phases # of non-Mar. Error
Poisson process 1 0/36 0.3569/0.1396
Scaled service process 2 0/36 0.1477/0.0961
Busy period based 12 0/36 0.119/0.219
ETAQA, truncation level=2 18 36/36 0.1364/0.068
ETAQA, truncation level=50 306 36/36 0.0034/0.007
Lag-1,4moments, Algorithm3 2 0/36 0.1116/0.0695
Lag-1,4moments, Algorithm5 2 0/36 0.1116/0.0695
Lag-1,4moments, Algorithm4 2 0/36 0.1116/0.0695
Lag-1,9moments, Algorithm3 3 12/36 0.076/0.0511
Lag-1,9moments, Algorithm5 3 0/36 0.076/0.0511
Lag-1,9moments, Algorithm4 3 0/36 0.076/0.0511
Table 7.: Properties of the departure process approximations for the tandem example with LBL input
8.2 single-type qeueing networks 135
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 10 20
Utilization of both nodes
Meannumberofcustomers
Simulation Lag-1based + alg.3 Lag-1based + alg.4 Lag-1based + alg.5 Busy period based
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0
1 2
Utilization of both nodes
Meannumberofcustomers
Simulation Lag-1based + alg.3 Lag-1based + alg.4 Lag-1based + alg.5 Busy period based
Figure 39.: Mean number of customers at node2of the tandem network with negatively correlated input
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 2 4 6 8
Utilization of both nodes
SCVofthenumberofcustomers
Simulation Lag-1based + alg.3 Lag-1based + alg.4 Lag-1based + alg.5 Busy period based
Figure 40.: Squared coefficient of variation of the number of customers at node2of the tandem network with negatively correlated input
Approximation algorithm # of phases # of non-Mar. Error
Busy period based 14 0/36 0.2654/0.2686
ETAQA, truncation level=2 24 36/36 0.0567/0.0303
ETAQA, truncation level=50 408 36/36 0.0011/0.002
Lag-1,4moments, Algorithm3 2 18/36 –/–
Lag-1,4moments, Algorithm5 2 0/36 0.0413/0.0566
Lag-1,4moments, Algorithm4 2 0/36 0.0411/0.052
Lag-1,9moments, Algorithm3 3 36/36 –/–
Lag-1,9moments, Algorithm5 3 0/36 0.0513/0.0615
Lag-1,9moments, Algorithm4 3 0/36 0.037/0.0596
Table 8.: Properties of the departure process approximations for the tandem example with negatively correlated input
136 qeueing network analysis based on the joint moments
Figure 41.: A more complex queueing network with 4 nodes