TRANSIENT ANALYSIS OF A QUEUE WITH QUEUE-LENGTH DEPENDENT MAP AND ITS APPLICATION TO SS 7 NETWORK

We analyze the transient behavior of a Markovian arrival queue with congestion control based on a double of thresholds, where the arrival process is a queue-length dependent Markovian arrival process. We consider Markov chain embedded at arrival epochs and derive the one-step transition probabilities. From these results, we obtain the mean delay and the loss probability of the nth arrival packet. Before we study this complex model, first we give a transient analysis of an MAP/M/1 queueing system without congestion control at arrival epochs. We apply our result to a signaling system No. 7 network with a congestion control based on thresholds.


Introduction
Congestion control based on thresholds [4, 7-10, 15] is aimed to control the traffic causing overload before a significant delay builds up in the network and so to satisfy the quality of service (Qos) requirements of the different classes of traffic.The QoS requirements are often determined by two parameters; the loss probability and the mean delay [5].S.Q.Li [10] proposed a congestion control with double thresholds consisting of an abatement threshold and an onset threshold to regulate the input rate according to the congestion status.Packets are classified as one of the two priori- ties: high priority and low priority.When the queue length exceeds the onset thres- hold, the low priority packets are blocked and lost until the queue length decreases to the abatement threshold.O.C. Ibe and J. Keilson [8] extended this model to the sys- tem with N doubles of thresholds (N _> 1) and N different priority packets.For the above systems, they assumed that the arrival processes are a Poisson process [8] and a Markov modulated Poisson process (MMPP) [10], and they obtained the steady state characteristics.
In order to find out performance of a congestion control, first we need to analyze the transient behavior of the system.The Laplace transform and z-transform methods [1, 6, 14, 16] are usually applied within conventional transient analysis.It seems not to be easy to analyze the transient behavior of the system with finite buffer and congestion control based on thresholds by the above transform methods.For transient analysis of such a system, D.S. Lee and S.Q.Li [11,12] used a discrete time analysis with its time indexed by packet arrivals.They assumed the arrival processes are MMPP [11] and a switched Poisson arrival process [12] and obtained the one-step transition probabilities of an embedded Markov chain.But they considered a conges- tion control with only one threshold called partial buffer sharing policy.
In this paper, we consider the congestion control with double thresholds as in [9] and [10].We assume that the arrival process is a queue-length dependent Markovian arrival process (MAP).The motivation of this model comes from the study of the congestion control in a signaling system No. 7 (SS7) network [15].A congestion con- trol called inernational control in a SS7 network is a reactive control with double thresholds, which uses a notification to inform senders about the congestion status of the system.Each sender regulates its traffic load to the system when he receives a notification, and uses timers to resume its traffic load.For such a system, the arrival process can be modeled as a queue-length dependent Markovian arrival process (MAP) [4].
For a transient analysis, we use the discrete time analysis as in [11,12] but using the advantage of simple notations of MAPs we obtain the one-step transition probabi- lities by a simpler derivation than that of D.S. Lee and S.Q.Li in [11,12].The models dealt with in [11,12] are special cases of our model.We obtain the mean delay and loss probability of the nth arrival packets.In order to evaluate the per- formance measures we give an algorithm, which enables us to reduce the complexity of iterated Kolmogorov equation in Section 3. We apply our result to analyze the international control in SS7 networks.In the numerical examples, we show the im- pact of various parameters such as the value of the thresholds and the length of times and input rates on the transient performances.
by using matrix formulation.In Section 3, we consider the congestion control based on double thresholds with queue-length dependent MAP and derive the transient queue length probability at arrival epochs.We give performance measures and an algorithm to compute the performance measures.In Section 4, we describe an analy- tic modeling of the international control in SS7 networks and present numerical exam- ples and observations.
2. Transient Analysis of MAP/M/1 Queueing System at Arrival Epochs In this section we study an MAP/M/1 queueing system without any congestion con- trol to provide a better understanding of the system with congestion control in Section 3. We assume that the packets arrive according to a Markovian arrival process (MAP) with representation (C,D), where C and D are m x m matrices [2, 3, la].
Here C is the rate matrix of state transitions without an arrival and D is the rate matrix of state transitions with an arrival.The service time of a packet is assumed to be exponentially distributed with parameter #.We denote the number of packets in the system and the state of the underlying Markov chain of the MAP at time by N(t) and J(t) (1 <_ J(t) <_ m), respectively.Then the two-dimensional process X(t) (N(t),J(t)) forms a continuous time Markov chain.Let T, denote the nth packet arrival epoch.Then we form an embedded Markov chain {X n In >_ 0} defined by Xn-(N(Tn+),J(Tn+)).
Let the nth step transition probabilities from X 0 to X n be denoted by pn.

One-Step Transition Probabilities
For convenience, we define the set of all states (i, j), 1 _< j < m, by level i.
Then we have the following lemma.
Lemmal" For o>1, Note that the left-hand side of (2) is the expected time that X(t) spends in state (io, k until the Markov chain departs from its level o starting from state (io, Jo)" Proof: See Appendix 5.1.
Proof." See Appendix 5.2.Now we are ready to arrive at the one-step transition probabilities.

Special Cases
From Proposition 3, we can obtain the following corollaries, which agree with the results in [11,12].Corollary 4: (Corollary 3 in [11]) If the arrival process is an MMPP with the re- presentation (Q,A), where Q is the infinitesimal generator of the underlying Markov chain, and A is the arrival rate matrix, then for <_ 0 + 1, Proof: An MMPP with the representation (Q,A) is an MAP with the representa- tion (C,D), where C=Q-A, D=A.Therefore, the result follows from (3) and Proposition 3 directly.
Corollary 5: (Proposition 2.1 in [12]) If the arrival process is a switched Poisson process with the representation (Q,A), where r I --rl] 0 "1 then for <_i o+1 and > l, and Zl,Z 2 are the tools of the quadratic equation #+r 1 +A # is diagonalizable with eigenvalues z 1 and z 2 which are roots of the quadratic equation whose columns are the eigenvectors of I -gQ +-}A with respect to z and z2, respect- ively.Then by simple calculation, we can obtain z o 0 For 1, we can obtain similarly ( ) This agrees with the result of Proposition 2.2 in [12].
3. Transient Analysis of the Congestion Control with Double Threshold In this section we consider a congestion control with double thresholds consisting of an onset threshold M and an abatement threshold L (see Figure 1).
Figure 1: The model of the congestion control with two thresholds Let B denote the buffer size.Until the queue length reaches the threshold M, the con- gestion status of the buffer is assigned to 0. Once the queue length exceeds M from below, we assign the congestion status of the buffer to 1 during the period until the queue length crosses L from above.When the queue length crosses L from above, the congestion status of the buffer is assigned to 0 again and the procedure is repeated.
We assume that the packets arrive according to an MAP with the representation (Co, Do) when the congestion status of the buffer is 0 and the packets arrive accord- ing to an MAP with the representation (C1,D1) when the congestion status of the buffer is 1 and Co, Do, C and D 1 are m x m matrices.We will describe the matrices Co, Do, C 1 and D 1 in detail for modeling of congestion control in SS7 networks in Sec- tion 4.
Let I(t) denote the congestion status of the buffer at time t.Then X(t)= (I(t), N(t),J(t)) forms a continuous time Markov chain and X n (I(T n + ), N(T n + ), J(Tn+)) forms an embedded Markov chain of the Markov chain {X(t) lt >_ 0}, where T n is the nth packet arrival epoch.Let the nth step transition probabilities be denoted by Pn(0, i0; , i) which is an m x m matrix, where [Pn(0' i0; ' i)]jo' J P{Xn (' i, j) Xo (0' io, Jo))" Let the one-step transition probabilities of X, be denoted by P(o, io; , i) PI(o, io; , i).
The proof is identical to that of Proposition 3 and is therefore omitted.

Performance Measures and Algorithm
Once the transient queue length probability Pn(o, io;,i at arrival epochs is obtained, the performance measures can be easily evaluated.Let d n be the delay of J0th the nth arrival packet and ej0 (0,...,0, 1 ,0,...,0).Under the initial condition Xo-(o, io, Jo), both the mean and the variance of d n are obtained as 1 E{d n Xo (o, io, Jo} Transient Analysis of a Queue with Queue-Length Dependent MAP 381 1 #2 min(i 0 + n,B) i-L+1 (i 1)2.ejoPn(o, io; 1, i)e -[E{d n Xo ((o, io, jo)}]2, (12) where e is the column vector whose elements are all 1.Let Plss(O, io, Jo) denote the loss probability of the nth arrival packet under the initial condition X o -(o, io, Jo).
Then n pn 1(o, io; 1, B)Gle.Ploss(o, io, Jo) When the matrices R o and 1 1 are diagonalizable, we can greatly reduce the com- plexity of the iterated Kolmogorov equations ( 9) and (10) using their eigenvalues and eigenvectors as in [11, 12].But there is no evidence that there exist distinct and real eigenvalues of the matrices R o and R1, and it is not easy to obtain the eigenvectors numerically.Therefore, we introduce another algorithm to reduce the complexity of the iterative Kolmogorov equations ( 9) and (10).To obtain the performance mea- sures, we only need to calculate the column vectors Pn(o, io;,i)e in ( 11) and ( 2), and pn-l(o, io;1,B)Gle in (13).

Application to SS7 Network
There are three types of congestion controls in SS7 networks such as international control, national option with congestion priorities, and national option without congestion priorities [15].Here we will describe the international control, when a message signal unit (MSU) is received at a Signaling Transfer Point (STP) for the congested link whose congestion status is 1, it is passed to Level 2 for transmission and a Transfer Controlled (TFC) packet is sent back to the originating Signaling Point (SP) which sent the MSU, for the initial packet and for every m 0 packet (default value of m 0 is 1 in this paper, but by a simple modification we can deal with the model with m 0 larger than 1).
We assume there are S identical SPs sending packets to a STP and we consider an output buffer of the STP and the packets sent to the output buffer (see Figure 2).If an SP receives a TFC packet from the STP, the traffic load toward the STP is reduced by one step, and two timers T29 and T30 are started where the length of T30 is greater than that of T29.Until T29 times out further TFC packets are ignored in order not to reduce traffic too rapidly.If a TFC is received after the expiry of T29 but before T30 expires, the traffic load is reduced by one more step and both T29 and T30 are restarted.This reduction continues until the last step when maximum reduction is obtained.If T30 expires, then the traffic load is increased by one step and Ta0 is restarted.This is repeated until the full load has been resumed.For simplicity, we assume the length of T29 to be equal to zero.The extension to the model with nonzero T29 is similar to the modeling in [4].Even though the lengths of Tao is deterministic, we assume that the length of T30 has an exponential distribution with a mean, which is a deterministic value for the analytical modeling as in [9].Letr--E[T30]" Let K be the maximum reduction step of the traffic load in an SP.Define the state of an SP as k (0 _< k _< K) if the SP ha8 reduced it8 traffic load k time8 since the beginning with full load.Assume that each SP whose state is k 8ends packet8 according to a Poisson process with rate $k (0 >--1 ----"K)" Let Yk(t) be the number of Sen in state k at time t.When (Y(t),Y2(t),...,YK(t))-(Yl,Y2,'",YK), s Z k =,lYk 8 the number of SP8 in state 0 and the total arrival rate to the STP i8 0( S E = lYk) --K E k 1AkYk" Hence, J(t) (Yl(t), Y2(t),..., YK(t)) governs the arrival rate and so it can be defined as the underlying process of the arrival to the STP with the state space consisting of (Yl,'",YK) listed in the lexicographic order, where Yi > 0 and l Yk < S. The total number rn of the states equals K!S! Let A((Yl,...,yK) (Yl,'",Y'K)) denote the transition rates from the state (Y, Y2,"', YK) to the state (y, Y2,'", YX) which are the elements of an m x rn matrix A. Let y-(Yl,'",YK)" For example, A(y,y) denotes the diagonal elements of matrix A. Let e be a vector whose elements are all zero except for the ith element ith which is 1, i.e., e i-(0,...,1,...,0).Let e 0-(0,...,0) and e K+I-eK, for the sake of convenience.We are ready to find the rate matrices C0, D0, C 1 and D 1 of the underlying process {g(t):t >_ 0} for our model in this section.Independently of the congestion status of the buffer of the STP, if a T30 of an SP whose state is k expires, the state of the SP will be changed to k-1.Hence, Cn(Y, Y ek + ek 1) or.Yk for n O, 1, 1 _< k _< K.
When the congestion status of the buffer of the STP is 0, there is no transition of the state with an arrival of the underlying process J(t), since there is no TFC generating from STP.
When the congestion status of the buffer of the STP is 1, each SP who sends a packet to the STP will receive a TFC and reduce the traffic load by one step and restart T30.Therefore, if one of yk's SPs whose states are k sends a packet to the STP, its state will be changed into k + 1.
The diagonal elements of the matrices C O and C are negative values to make C0e + Doe 0 and Ce + Die 0, respectively.The elements of the matrices Co, Do, C and D1 not mentioned above are all zeros.
For all numerical examples, we assume that S-10, K-1 and that the time scale is normalized by the mean service time of a packet, i.e., #-1.0.Let the buffer capacity B be equal to 50.For Figures 3 through Figure 6, we assume A 0 0.08 and A 1 0.04. Figure 3, Figure 4, and Figure 5 display the mean delay and the loss probability of packets in terms of functions of time, when T30-100.For Figure 3, we let L be fixed at 25 and the initial state by (0, 25, 0).As M decreases, the congestion control is triggered earlier and therefore the mean delay and the loss probability of packets decrease as shown in Figure 3. Figure 3.The mean delay and the loss probability of packets for the case of T30-100,L-25 and that the initial state equals to (0, 25, 0) In Figure 4, we consider an epoch when the queue length exceeds the onset threshold M as the initial epoch, i.e., X 0 (1, M + 1, 1).Since the congestion control is triggered from the initial epoch, each SP receives a TFC packet when it sends a packet until the queue length crosses the abatement threshold L. SPs receiving a TFC packet reduce their traffic load and therefore the total offered load until the buffer decreases and the mean delay of packets decreases as shown in Figure 4.The loss probability of packets increases initially but it begins to decrease soon since the congestion control is triggered.After a time interval, the mean delay and the loss probability increase slightly as in Figure 4.This is because the total offered load in- creases again after the queue length crosses L. Figure 4.The mean delay and the loss probability of packets for the case of T30-100, M 40 and that the initial state equals to (1,M + 1, 1) B.D. CHOI, S.H. CHOI, D.K. SUNG, T.H. LEE and K.S. SONG In Figure 5, we compare the loss probabilities with different M and different initial state X 0 for a fixed L (L-25). Figure 5 shows that the loss probabilities converge to the same value for the same M independently of the initial value as the time increases. . . . .iO 31 .--....-_____.'_'_.:'.'7.'.-.'7-':--'---'---"-"'-":------- Figure 5.The mean delay and the loss probability of packets for the case of T30-100, L-25 Define Fn(io, Jo,o) as the mean number of SPs, which send packets in their full traffic load at time n (in packets).Then Fn(io, Jo,o)is calculated by PU(o, io;,i as Fn(io, Jo, o) S ejoPn((o, io; (, i)e*, where e* (0, 1,2,..., S).As in the case of plrss(io, Jo, (o), Pn((0, i0; (, i)e* can be evaluated iteratively by substituting pn-l((o, io;(,i)e and Pn((o, io;(,i)e for pn-1((o, io; (,i)e* and pn-1((o, io; (, i)e* in (14), respectively.Figure 6 displays Fn(O,L,O) and Fn(1,M + 1,1) in terms of functions of time.For a fixed L, as M decreases and for a fixed M, as L decreases, the mean number of SPs with full traffic load decreases as in Figure 6.Hence there is trade-off between the loss probability (and the mean delay) and the throughput in terms of Fn(io, Jo, o)" Figure 6.The mean number of SPs with their full traffic load for the case T30-100, L 25, M 40 and that the initial states equal to (0, L, 0) and (1, M + 1, 1), respectively.
In Figures 7 and 8, we consider the control of SP with traffic reduction and the timer.We consider the case of M-40 and L-25. Figure 7 shows the impact of the length of the timer T30 on the transient performance.As T30 increases, the time of resuming full traffic load of each SP is delayed and therefore the loss probability and the mean delay of packets decrease as in Figure 7.We assume that each SP d receiving TFC only send d% of its full packets.Then A 1 -]-0--6A0.We consider two cases: A0 0.08 and A0 0.12.As d decreases, the total offered load decreases and therefore the mean delay decreases as shown in Figure 8. Figure 7.The mean delay and the loss probability of packets for the case of L 25, M 40 and that the initial state equal to (1, M + 1, 1) Transient Analysis of a Figure 8.The mean delay of packets for the case of L 25, M 40 and that the initial state equal to (1, M + 1, 1) with "o 0.08 and 'o 0.12, respectively Manfield, D.R., Millsteed, G. and Zukerman, M., Congestion controls in SS7 signaling networks, IEEE Commun.Mag.June (1993), 50-57.Sharma, O.P. and Shobha, B., Transient behavior of a double-channel Markov- ian queue with limiting waiting space, Queueing Systems 3 (1988), 89-96.