Performance Analysis of Wavelength Division Multiplexing Asynchronous Internet Router Employing Space Priority Mechanism under Self-Similar Traffic Input — Multi-Server Queueing System with Markovian Input and Erlang-k Services

In this paper, we analyze the queueing behaviour of wavelength division multiplexing (WDM) Internet router employing partial buffer sharing (PBS) mechanism with self-similar traffic input. In view of WDM technology in networking, each output port of the router is modelled as multi-server queueing system. To guarantee the quality of service (QoS) in Broadband integrated services digital network (B-ISDN), PBS mechanism is a promising one. As Markov modulated Poisson process (MMPP) emulates self-similar Internet traffic, we can use MMPP as input process of queueing system to investigate queueing behaviour of the router. In general, as network traffic is asynchronous (unslotted) and of variable packet lengths, service times (packet lengths) are assumed to follow Erlang-k distribution. Since, the said distribution is relatively general compared to deterministic and exponential. Hence, specific output port of the router is modelled as MMPP/Ek/s/C queueing system. The long-term performance measures namely high priority and low priority packet loss probabilities and the short-term performance measures namely mean lengths of critical and noncritical periods against the system parameters and traffic parameters are computed by means of matrix-geometric methods and approximate Markovian model. This kind of analysis is useful in dimensioning the router under self-similar traffic input employing PBS mechanism to provide differentiated services (DiffServ) and QoS guarantee. How to cite this paper: Gudimalla, R.K. and Perati, M.R. (2016) Performance Analysis of Wavelength Division Multiplexing Asynchronous Internet Router Employing Space Priority Mechanism under Self-Similar Traffic Input—Multi-Server Queueing System with Markovian Input and Erlang-k Services. Applied Mathematics, 7, 1707-1725. http://dx.doi.org/10.4236/am.2016.715144 Received: July 20, 2016 Accepted: September 13, 2016 Published: September 16, 2016 Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access R. K. Gudimalla, M. R. Perati 1708


Introduction
It is evident from seminal studies that Internet protocol (IP) traffic of both Ethernet traffic and wide area network (WAN) traffic exhibits self-similarity [1]- [3].Markov modulated Poisson process (MMPP) is employed to emulate the self-similar traffic over the different time scales [4]- [6].Naturally, high demand in Internet traffic leads to congestion problems.Congestion problems can be dealt with some space priority mechanisms.Buffer Access Control (BAC) mechanism is one of the space priority mechanisms.There are several strategies to implement this mechanism and one of such strategies is partial buffer sharing (PBS) mechanism.In this scheme [7]- [11], there is a limit (or threshold) in the buffer, and the part of buffer on or below the limit is shared by all arriving packets.When the buffer occupancy is above the limit, the arriving low priority packets will be dropped, and only high priority packets will be allowed.High priority packets will be lost only when buffer is full.If the limit is relatively low, then more low priority packets will be lost.If the limit is relatively high, then more high priority packets will be lost.The limit setting induces two time periods, namely, critical and non-critical periods.The critical period is the time period during which buffer occupancy of the queueing system is on or above limit level and the non-critical period is the time period during which buffer occupancy is below the limit level.This way, there is a trade-off relation between limit setting and packet loss.Hence, optimum level of limit is very important in dimensioning the network nodes such as switches or routers.
Priority queue models are briefly discussed below.In paper [12], the queueing analysis of infinite buffer priority system with MMPP input is investigated with an assumption that the delay sensitive cells and non-delay sensitive cells arrive at two separate queues.This scheme is not realistic as the buffers consist of limited number of fiber delay lines (FDLs) with fixed granularity.The loss behaviour of finite buffer space priority queues with discrete batch Markovian arrival process (D-BMAP) has been analyzed [7] which is not the case, since the router under consideration is handling self-similar traffic which is modelled by continuous-time Markov process.
Another issue of Internet traffic is to provide Quality of Service (QoS).Internet router with wavelength division multiplexing (WDM) technology is promising one to guarantee QoS.In WDM router, there are N input fiber lines, N output fiber lines, and each fiber line has C wavelength channels and a wavelength converter pool of size s, ( ) dedicated to each output fiber line.Hence, each output port of the router is to be modelled as multi-server queueing system [13] [14] [11].In general, there are two types of networks: synchronous (slotted) and asynchronous (unslotted) [15].In the case of first one, all the packets are of constant size [7] [13] [9] [12].In asynchronous networks, all the packets have variable lengths [8] [14] [16] [11].Since IP packets are, in general, variable in length, the router is required to possess the ability to router the variable length packets.Therefore, performance analysis of the router by means of MMPP/D/1/C queueing system wherein service time (packet length) is deterministic may not be appropriate [17].Router with variable length packet traffic is modelled as MMPP/M/1/C queueing system wherein service time is exponential distribution [18]- [20].In the papers [8] [13] [14] [11], the router is modelled as either single-server or multi-server priority based queueing system.However, the service times (packet lengths) distribution is assumed to be deterministic or exponential.the high priority packets are for the whole buffer space.The threshold setting induces two time periods, namely, non-critical period and critical period [7] [8].Non-critical period is the time period during which buffer occupancy is below the threshold level and critical period is the time period during which buffer occupancy is on or above threshold.Each priority traffic is self-similar and is modelled by MMPP process [6].

WDM Asynchronous
Assume that high priority (class 1) packets and low priority (class 2) packets arrive at the system according to MMPPs of the states 1 m and 2 m , respectively, and are go- verned by the matrices process is in state i, the next departure of class p occurs no later than time t with the process in state j, during the service time there are m packets.We consider the Markov , , / where n L denotes the buf- fer occupancy and n J denotes the phase of superposed MMPP.For convenience, a queueing system is said to be at level d, if buffer occupancy is equal to d (excluding the ones in service).We ignore the time spent in a state and consider only the number of packets arrived during the sojourn time.Therefore, pertinent system is embedded Markov chain and has the following irreducible transition probability matrix P (with the dimension ( ) ( ) : where In Equations ( 2) and ( 3), the elements of row and column outside the matrices 1 P and 2 P are state spaces of the Markov chain and the elements of first ( 1 s + ) rows are identical. ( ) A and ( ) 2 , 0 i A i ≥ are the matrices of high priority and low priority packets, respectively.The overall input traffic is the superposition of high priority and low priority packets, which is also an MMPP with , Sojourn time is ignores the matrices of counting functions become independent of time t.The matrices ( ) A satisfies the following equation [18]- [20], If the service time distribution

( )
H t is Erlang with k phases in series of service and mean service time , where µ is the mean service rate.We have dropped the superscript "p" for convenience as the procedure holds good for both high priority and low priority packets.The matrices of counting function m A 's can be computed following the procedure [18]- [21].Then Equation (4) reduces to where I is the unit matrix of appropriate dimension.For 0 m = in Equation ( 5), we have For , 1 , n th term of the series on right hand side of Equation ( 5).Then we have ( ) Now, multiply on both sides of above equation by 1 w n In above equation, equating the coefficients of like powers of z, we obtain, ( ) ( ) Equating the coefficients of n z in the Equation ( 5), we get ( ) The fundamental arrival rate of class p packets is ( ) ( ) ( ) , where ( ) p π is the steady state probability vector of ( ) , , , C s y y y y − =  , where is the conditional probability that there are v packets in the system given that embedded Markov chain is in the ( ) , l m state.Therefore, we have , 1 yP y ye = = , where e is the column vector consisting of all 1 [21]- [24].Then, in the steady state, high priority packet loss probability hp P and low priority packet loss probability lp P , respectively, are derived as follows [13] [7].Let hp PL denote the number of high priority packets lost due to the fact that buffer is full.Then the expected value of  and is obtained by considering the last column of the ( ) ( ) block transition probability matrix P as that column consists of matrices containing conditional probabilities that the buffer is full.Then high priority packet loss probability hp P is given by where ( ) ( )  is the number of packet arrivals during the mean service time.Let lp PL denote the number of low priority packets lost due to the fact that buffer occu- pancy exceeds the threshold.Then expected value of  and is obtained by considering the last ( ) b s + columns of the ( ) ( ) probability matrix P as these columns consists of matrices containing conditional probabilities that the buffer occupancy is greater than or equal to threshold.The low priority packet loss probability lp P is given by ) In the paper [7], the alternate methods to compute hp P and lp P are proposed, but the input process is assumed to be discrete batch Markovian arrival process (D-BMAP) and the pertinent queueing system is of single-server.On the same lines we have extended it to multi-server queueing system with continuous input process.In view of threshold setting we decompose the state space U into two subsets: and This partition of U makes the transition probability matrix P decomposed as follows: , * , nc nc c c nc c The sub-matrices nc P , Similarly, in the case of critical period, the TPM of the absorbing Markov chain that has transient states c U and absorbing states nc U is given by , 0 c c nc c where , sm m columns of the matrix * , c nc P .This is followed from the fact that for a critical period it suffices to attain the buffer level The absorbing probability vectors are given by [ ]( ) , respectively, where V is the ( ) ( ) The average total number of high priority packets lost during a critical period is The average total number of low priority packets lost during a critical period is The high priority packet loss probability hp P and low priority packet loss probability lp P are and

Computational Complexity
In this section, first we compute the computational complexity of the long-term performance measures, namely, high priority packet loss probability hp P and low priority packet loss probability lp P through the Equations ( 8)- (9).Next, we compute the short-term performance measures, namely, the mean length of non-critical periods  15)-( 26) and analyze their computational complexity [7].
In order to find the computational complexity of long-term hp P and lp P through the Equations (( 8)-( 9)), the transition probability matrix P in Equation ( 1) is not of the canonical / /1 M G type, using the Schur-Banachiewicz inversion formula, to compute the steady-state probability vector y of P (with dimension ( ) ( ) , where 1 P is the matrix P in which the last column is replaced by the column vector [ ] T 1, 1, , 1, 0 − − −  .Let the permutation matrix M with dimension ( ) ( ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 where 0 and I are the zero and the identity matrices of dimension 1 2 m m .Now, multip- lying M to ( ) The sub-matrices E, F, G, and H are of the dimensions ( ) ( ) , and sm m sm m × , respectively.Then the steady state probability vector y is the last row of the matrix, ( ) By the Schur-Banachiewicz inversion formula, where sm m sm m × .But E is not a Toeplitz matrix and can be further decomposed as where 11 E and 22 E are upper-triangular and Toeplitz matrices of dimensions bm m , respectively.Thus The sub-matrices  ( ) . Thus the overall complexity to compute hp P and lp P by Equations (( 8)-( 9)) is of the order ( )( ) ( ) , a complexity equal to the inversion of ( ) E L , hp P , and lp P and analyze their computational complexity [7].
The computational complexity of the steps 5-6 in the above algorithm is of the order ( ) ( ) due to the products of several pairs of row and column vectors.
The computational complexity of the steps 3-4 is of the order We next show that the computational complexity of the first step is of the order ( ) ( ) by subject to the existence of the inverses of 0 A , ( ) The sub-matrices nc E , nc F , nc G , and sm m rows of ( ) and we have ( ) The sub-matrices c E , c F , c G , and c H are of the dimensions sm m bm m × , and sm m sm m × , respectively.We have ) where sm m sm m × .Since c E is upper-triangular Toeplitz block matrix, the inverse is also upper-triangular Toeplitz block matrix.The computation complexity of obtaining the inverse of ( )

Numerical Results
Using the Equations ( 15)-( 26), the steady state packet loss probabilities and mean length of non-critical and critical periods are computed [7].The generalized variance based Markovian fitting method proposed in [6] is employed to emulate the self-similar traffic for both priority packet traffics.The mean arrival rate ( λ ) and variance ( 2 σ ) of the self-similar traffic is set to be 1 and 0.6, respectively [6], the interested time-scale range to emulate self-similarity is over , and the results are presented in Figures 2-8. Figure 2, depict the high priority packet loss probability decrease and the low priority packet loss probability increase as threshold (b) increases.In order to find out the optimal level of the threshold, we illustrate a plot of the high priority packet loss probability against the low priority packet loss probability ones at various b in Figure 3.We could find out that the optimal level of the threshold is the one located nearest to the left lower corner     5 depict the variation of packet loss probability against traffic intensity (rho) and buffer capacity, respectively.It is clear that packet loss probability increase as traffic intensity increases (Figure 4).We observe that of high priority and low priority packet loss probabilities both decrease as buffer capacity increases (Figure 5).From Figure 6, we observe that the mean lengths of non-critical periods (ELNC) and critical periods (ELC) are decreases as threshold increases.Figure 7 & Figure 8 depict the variation of mean lengths at the optimum level of threshold against traffic intensity and buffer capacity, respectively.Figure 7 illustrates the mean lengths of non-critical periods decrease and critical periods increase   as traffic intensity increases.Figure 8 depict the mean lengths of non-critical and critical periods both increase as buffer capacity increases.

Conclusion
In this paper, we have investigated the performance of asynchronous router employing PBS mechanism to provide differentiated services under Markovian modelled self-similar traffic input.To reduce the computational complexity, the original high dimensional MMPP of the low priority packets is approximated by 2-state MAP.The long-term performance measures, namely, the steady state high priority and low priority packet loss probabilities, and the short-term performance measures, namely, average length of non-critical and critical periods, are computed and presented graphically.With this analysis, we can locate the optimal limit (threshold) position of buffer to obtain the greatest performance.
Router-Multi-Server Queueing Model MMPP/Ek/s/C Employing Partial Buffer Sharing (PBS) Mechanism with Erlang-k Service Times We consider the WDM asynchronous N N × router with each output fiber line consisting of C wavelength channels and a wavelength converter pool of size s.Buffer depth then is C s − .Such a router with self-similar traffic input can be modelled as MMPP/E k /s/C queueing system.The operation and multi-server queueing model of the router employing PBS mechanism is shown in Figure 1.For the simplicity, two priority traffics are considered.The threshold is set at the level 2 1 C s b − − + , where b is a posi- tive integer.The low priority packets can only access first 2 C s b − − buffer spaces and

=Figure 1 .
Figure 1.Operation and multi-server queueing model of a specific output port of the router employing partial buffer sharing mechanism with two different priority traffic input and Erlang-k service times.
the departure epochs of the queueing system on the state space c P are the left upper part, right upper part, left lower part, and right lower part of the matrix P with dimensions of ra- ther than all other below levels which is the starting point of non-critical period of each cycle except the first one.The absorbing probability vectors 2 C s b α − − and β of the Markovian chain ncP and c P described in the paper[7] are given by, average length of non-critical and critical periods are [ ]

PP
are the high priority packet loss during a critical period and , are the low priority packet loss during a critical period, and are given below, for 0 l > , ( and the long-term performance measures, namely, high priority packet loss probability hp P and low priority packet loss probability lp P through the Equations (

22 E
− are also upper-triangular and Toeplitz matrices and the existence of

22 E 2 A . The complexity to obtain 1 E
− is due to the existence of the inverses of 0 A , − is of the order the complexity to obtain the Schur complement X and the matrix R in (29) is of the order ( ) place the algorithmic steps needed for computing the performance meas- and β by using 16) in which the last column is replaced by the column vector [ ] nc E is upper-triangular Toeplitz block matrix, the inverse is also upper-triangular Toeplitz block matrix.Thus the computation complexity of obtaining nc S in (33) is of the order due to the computation of the inverse of nc E and the mul- tiplication of the matrices nc G and1 nc E − .Similarly, we have

.
In conclusion, the overall complexity of the algorithm is above for the computation of the performance measures

16 ≥. That is, both 1 m and 2 m must be 16 ≥ . So each class is here characterized by 16 16 ×
3].In the papers[5] [6] it is shown that in order to emulate self-similar traffic well, the minimum number of states of the resultant MMPPs must be matrices.Such a high dimensional MMPP for both high priority and low priority traffic results in computational complexity.In order to reduce the computational complexity, we use approximate model[9], which is based on the papers[25] [26].The resultant 16-state MMPP of low priority packets is approximated by a 2-state Markovian arrival process (MAP).By applying this approximated model, the computational complexity is reduced by3  8 512 = times[16].The number of servers (s) is set to be 3, the number of phases in series of each server (k) is set to be 5, and the system capacity (C) is set to be 20, buffer depth of the router then is values