Buffer Overflow Period in a MAP Queue

The buffer overflow period in a queue with Markovian arrival process
(MAP) and general service time distribution is investigated. The results include
distribution of the overflow period in transient and stationary regimes and the distribution of the number of cells lost during the overflow interval. All theorems are
illustrated via numerical calculations.


Introduction
One of the crucial performance issues of the single-server queue with finite buffer (waiting room) is losses, namely, customers (packets, cells, jobs) that were not allowed to enter the system due to the buffer overflow.This issue is especially important in the analysis of telecommunication networks.
Sample evolution of the queue length process in a finite-buffer system is depicted in Figure 1.1.In a time interval, where the number of customers in the system is equal to its capacity, all arrivals are blocked and lost.We call this interval buffer overflow period and it is equivalent to the remaining service time upon reaching a full buffer.Obviously, the duration of the buffer overflow period is responsible for the number of losses that occur consecutively.
Distribution of the buffer overflow period deserves our attention for many practical and theoretical reasons and these motivations can be found by the reader in the literature (e.g.[1,2]).Herein, we add just one more argument that makes the overflow interval an interesting subject to study.
The overflow interval plays an important role in the design of forward error correction (FEC) algorithms in packet networks.The FEC technique is based on adding redundant packets to the original sequence in order to protect the transmitted data from the loss of Buffer size Buffer overflow periods 1. Sample path of the queue size process in a single-server system with finite buffer.
packets [3].Precisely, to each original block of k consecutive packets, we add h redundancy packets, so that the loss of at most h packets in the resulting k + h block can be recovered.FEC is particularly useful in real-time audiovisual transmission, in which the retransmission of lost packets would take too much time [4].Naturally, distribution of the buffer overflow period has a direct impact on the number h that provides a low level of unrecoverable losses.Pay attention to the fact that the classic parameter, loss ratio, does not give sufficient information about the level of unrecoverable losses.This effect was also studied in [5].Using a simple M/G/1/b queue, it was shown there that by manipulating the model parameters we can keep the overall loss ratio on a constant level while obtaining drastically different distributions of consecutive losses.
To the best of the author's knowledge, there are no reported results on the duration of the buffer overflow period and the structure of consecutive losses in a queue with autocorrelated input rate.The previous works on the overflow period were concentrated on simple Poisson arrivals [1,[6][7][8], batch Poisson arrivals [2], or renewal arrivals [8,9].However, these processes are not able to mimic the complex autocorrelation structure of observed network traces, which can be bursty and self-similar [10,11].Naturally, these properties have a deep impact on the buffer overflow period.As the Markovian structure, up to some extent, can mimic the self-similar behaviour [12,13], the MAP and its special cases (MMPP) have been widely used in computer networks traffic modeling [14][15][16][17][18][19].Queues fed by a MAP are usually studied assuming an infinite buffer (see, for instance, [20][21][22][23][24] and the references given there).The number of papers dealing with characteristics of finite-buffer MAP systems is rather limited [25][26][27][28][29] and none of them is devoted to the overflow period.
The remaining part of the paper is organized in the following manner.In Section 2, a detailed description of the queueing model and notation used throughout the article are given.Section 3 presents the distribution of the first buffer overflow period with a proof, some auxiliary results, and comments.Distribution of the state of the underlying chain at the end of the overflow interval is computed in Section 4. Based on this result, the formulas for subsequent and stationary overflow periods are obtained.In Section 5, distribution Andrzej Chydzinski 3 of the number of consecutive losses that occur during the overflow period is presented.In Section 6, a set of results for the popular special case of the MAP arrival process, which is the Markov-modulated Poisson process (MMPP), is given.Two numerical examples, that illustrate the theorems proven, are presented in Section 7. In particular, in Example 7.2 a set of numerical results based on buffering of aggregated IP traffic is reported.Finally, remarks concluding the paper and propositions of future work are gathered in Section 8.

Queueing model and notation
Let N(t) denote the number of arrivals in (0, t] and J(t), 1 ≤ J(t) ≤ m, the state of the underlying Markov chain.Then the Markovian arrival process is defined as a 2-dimensional Markov process (N(t),J(t)) on the state space {(i, j) : i ≥ 0,1 ≤ j ≤ m} with an infinitesimal generator R in the form where D 0 and D 1 are m × m matrices.D 1 is nonnegative, D 0 has nonnegative off-diagonal elements and negative diagonal elements, and An alternative, constructive definition of a MAP is the following.Assume the underlying Markov process is in some state i, 1 ≤ i ≤ m.The sojourn time in that state has exponential distribution with parameter μ i .At the end of that time, there occurs a transition to another state and/or an arrival.Namely, with probability p i (0, j), 1 ≤ j ≤ m, there will be a transition to state j without arrival and with probability p i (1, j) there will be a transition to state j with an arrival.It is assumed that (2. 2) The following relations between parameters D k and μ i , p i (k, j) hold: We consider herein a single-server queueing system fed by a MAP.The service time is distributed according to a distribution function F(•), which is not further specified, and the standard independence assumptions are made.The buffer size (system capacity) is finite and equal to b (including service position).This means that if a cell (customer) at its arrival finds the buffer overflowed, it is blocked and lost.We assume also that the time origin corresponds to a departure epoch.In Kendall's notation, the system described is denoted by MAP/G/1/b.
The following notation will be used throughout the article: (i) J(t) is the state of the underlying Markov chain at the moment t, (ii) X(t) is the queue size at the moment t (including service position), (iii) P(•) is the probability, (iv) P i j (n,t) = P(N(t) = n,J(t) = j | N(0) = 0,J(0) = i) is the counting function for the MAP, (v) δ i j is the Kronecker symbol (δ i j = 1 if i = j and 0 otherwise), (vi) 1 is the column vector of 1's, (vii) the indicator function is (2.4)Moreover, the following m × m matrices will be of use: ) ) (In this notation [z i j ] i, j denotes an m × m matrix with elements z i j .)Finally, if {X k } ∞ k=1 is a sequence of m × m matrices (or column vectors of size m), we define an operator K b as follows: (2.9) Throughout the paper, it is assumed that matrices A 0 and G b are nonsingular.This assumption does not influence the applicability of the presented results-the author did not find any case with singular A 0 or G b during his numerical experiments.However, it seems to be hard to prove the nonsingularity in general (see also [26, page 281] for some remarks on the singularity of A 0 ).Andrzej Chydzinski 5

First overflow period
In this section, we are going to study the distribution of the first buffer overflow period (β 1 in Figure 1.1), depending on the initial queue length, X(0), and the initial state of the underlying chain, J(0).Formally, β 1 is defined as a difference ζ − τ, where τ is the first moment at which the buffer is full and ζ is the first departure moment after τ.
To present distribution of the first buffer overflow period, we will be using its tail: usually in a column-vector form and the distribution function (also in a column-vector form) Duration of the first buffer overflow period in the MAP/G/1/b system is distributed according to where and K b (g(t)) is defined in (2.9).(Naturally, replacing X k by g k (t) in (2.9) is required.) Before the proof is presented, let us make some observations.Firstly, it can be checked that if m = 1 then representation (3.4) reduces to the M/G/1/b-system formula (see [7,Theorem 3]).Secondly, vectors g k (t) can be computed by means of the uniformization technique (see [21, page 33]).Finally, to prove (3.4), the following lemma will be needed (see [30]).Lemma 3.2.Assume that A 0 , A 1 , A 2 ,... and Ψ 1 , Ψ 2 ,... are known sequences of m × m matrices and A 0 is nonsingular.Then every solution of the system of equations has the form where C is an m × m matrix which does not depend on n and the sequence R k is defined in (2.7).
Note that if Ψ 1 , Ψ 2 ,... is a sequence of column vectors of size m, then Y n and C also become column vectors and Lemma 3.2 remains valid.
Proof of Theorem 3.1.Utilizing the total probability formula with respect to the first departure moment, we have for every 0 < n < b, where (D 1 ) jk denotes an element of the matrix D 1 in the position ( j,k).The first term in (3.8) corresponds to the case where there is no buffer overflow before the first departure moment, u.Therefore the number of arrivals in (0,u) must be less than b − n.The second term corresponds to the case where at some moment v < u the buffer gets overflowed.In this case, we have β = u − v.If initially the system is empty, for every 1 ≤ i ≤ m we get Integrating the second term in (3.8) by parts and using matrix notation, we may rewrite (3.8) as while (3.9) gives ) Now, applying Lemma 3.2, the system (3.12) has the following solution: where c(t) is a column vector which does not depend on n.Putting n = 1 into (3.14),we get c(t) = A 0 ϕ 1 (t) and Finally, using (3.16) and the boundary condition (3.11) yields which finishes the proof of Theorem 3.1.
When applying formula (3.4), we usually store all matrices A k , R k and vectors g k .Therefore, a memory space for 2b m × m matrices and b vectors of size m is required.Regarding the time complexity, if we assume that m × m matrix multiplication and inversion are of O(m 3 ) order, then the number of floating-point operations needed to compute (3.4) grows as O(m 3 b 2 ).The main cost is connected with the computation of b matrices R k using (2.7).

Subsequent overflows and stationary distribution
In this section, we are going to study distributions of the subsequent overflow periods (β 2 , β 3 , ... in Figure 1.1) and the limiting distribution of β k as k → ∞.
If the queue is fed by a plain Poisson process, the solution is simple, as all of β k except β 1 have the same distribution.This is due to the fact that after the overflow period ends, the number in the system is b − 1, and this is the initial queue size for the next overflow period.Moreover, the distribution of β k , k ≥ 2 is equal to the distribution of β 1 assuming X(0) = b − 1.Thus in the case of a Poisson input, the formula for the first overflow period immediately gives the subsequent distributions and the stationary distribution, just by putting In the case of a queue fed by a MAP, β k has a different distribution for every k.Again, the initial queue size for every β k , k ≥ 2 is b − 1, but now the distribution of β k depends also on the state of the underlying Markov chain at the end of the previous overflow period.Therefore, we have to know how the state of this chain is distributed at the end of the overflow period.
For this purpose, let us consider a random sequence {α k } ∞ k=0 defined as follows: α 0 = J(0) and α k , k ≥ 1 is equal to the state of the underlying Markov chain at the end of the kth overflow period.Therefore the distribution of β 1 depends on α 0 and X(0), while the distribution of β k , k ≥ 2 depends only on α k−1 .It is easily seen that {α k } ∞ k=0 is a discretetime Markov chain. Denoting the following will be shown.
Theorem 4.1.Distribution of the state of the underlying Markov chain at the end of the first buffer overflow period has the form where and K b (U) is defined in (2.9).In particular, Conditioning on the first departure epoch, we obtain for every 0 < n < b, Again, the first term in (4.4) corresponds to the case where there is no buffer overflow before the first departure moment, u.The second term corresponds to the case where at some moment v < u, the buffer gets overflowed and in this case we have P(α The boundary condition has the form We can easily finish the proof by proceeding in the same manner as in the case of Theorem 3.1.Now, using Theorems 3.1 and 4.1, it poses no problem to obtain the distribution of β k for an arbitrary k.We only have to observe that S n is a transition matrix for the chain α k in the first step (α 0 → α 1 ) and S b−1 is a transition matrix in every subsequent step (α 1 → α 2 , α 2 → α 3 , etc.).We get the following.
Corollary 4.2.If k ≥ 2, then the duration of the kth buffer overflow period in the MAP/G/ 1/b system is distributed according to where vector v is a distribution of the underlying chain at t = 0, n is the queue size at t = 0, S n and H n (t) are given by (4.2) and (3.4), respectively.
Finally, as a consequence of (4.8), we obtain the following.
Corollary 4.3.Limiting distribution of β k as k → ∞ has the form where w is a stationary vector for the matrix S b−1 (wS b−1 = w, w1 = 1).
The existence and the uniqueness of the limiting distribution follows from the fact that the matrix S b−1 is irreducible and aperiodic.

Structure of consecutive losses
In this section, we are going to study the distribution of the number of losses that occur during the first, subsequent, and stationary overflow periods.
In the case of a Poisson arrival process, the probability that a group of i consecutive arrivals is lost during the overflow period is simply equal to ∞ 0 e −λt (λt) i i! dH(t). (5.1) In the case of a MAP arrival process, we cannot use any formula similar to (5.1).This is due to the fact that we do not know the state of the underlying chain at the beginning of the overflow period.Moreover, this state may be correlated with the duration of the overflow period.Therefore, we have to proceed in similar way as in Section 4.
If we denote by γ k the number of arrivals lost during the kth buffer overflow period and q n (l) = q n,1 (l),..., q n,m (l) T , ( then the following theorem can be proven.
Theorem 5.1.Distribution of the number of arrivals lost during the first buffer overflow period has the form with v k (l) = A k+l 1 and K b v(l) given in (2.9).
Proof of Theorem 5.1.Once again, conditioning on the first departure epoch, we obtain for an initially nonempty system (5.4) The boundary condition for an initially empty system is identical with (4.5), just replacing with gamma's is required.Thus we have and the remaining part of the proof is the same as in the case of Theorem 3.1.
Proceeding in the same manner as in the previous section, we may now obtain the number of losses in the kth overflow period as well as its limiting distribution.

Corollary 5.2. If k ≥ 2, then the distribution of the number of arrivals lost during the kth overflow period has the form
where vector v is a distribution of the underlying Markov chain at t = 0, n is the queue size at t = 0, S n and q n (l) are given by (4.2) and (5.3), respectively.Moreover, the number of arrivals lost during the overflow period in steady state is where w is a stationary vector for the matrix S b−1 .

Special case: MMPP
A Markov-modulated Poisson process is constructed by varying the arrival rate of the Poisson process according to an m-state continuous time Markov chain (see, for instance, [24]).Namely, when the Markov chain is in state i, arrivals occur according to a Poisson process of rate λ i .Therefore, MMPP is parameterized by two m × m matrices: (i) Q is infinitesimal of the continuous time Markov chain, (ii) Λ = diag(λ 1 ,...,λ m )-its diagonal elements are equal to arrival rates, nondiagonal elements are zeroes.It is easy to check that the MMPP is the special case of a MAP with the following parameters: Therefore, all the theorems proven above are valid also for the MMPP, only a slight change in notation is required.In particular, we have now (6.2)

Numerical results
Example 7.1.Let us start with the same MMPP parameterization that was used in [25].Namely, we assume the constant service time as a time unit and The stationary distribution for Q is then π = (0.00589,0.99411)and the load offered to the queue is around 49%. Basic steady-state parameters of this system, like distribution of the queue size and workload, can be found in [25].Herein, we are going to examine overflow intervals and structure of consecutive losses of this system.For visualization, the probability density function will be used rather than the distribution function H n (t). Figure 7.1 shows the dependence of the density of the first overflow period on the buffer size.In part (a), the initial state of the underlying chain is 1, while in part (b) it is 2. In both parts, the queue is initially empty.We may observe that as b grows, the distribution of the first overflow period converges quickly to some limiting distribution.This is also confirmed in the plot representing average values (Figure 7.2.).The average duration of the buffer overflow stabilizes for b around 5 or 13, depending if the initial state of the underlying chain is 1 or 2. It is also interesting that the average value may not change monotonically (Figure 7.2(b)).
In Figure 7.3, the dependence of the density of the first overflow period on the initial queue size is depicted.For a wide range of n the density function does not change significantly, but it changes rapidly, as n closely approaches b.The stationary vector for S b−1 is then w = (0.998437,0.001563).Pay attention to the different values of π and w.While in general the underlying chain stays in state 2 for the majority of time, the state at the end of the overflow period is usually 1.This can be explained by the fact that the overflow usually occurs when the state of the underlying chain is 1 (λ 1 is higher than λ 2 ), and this state remains to the end of the overflow interval.
Figure 7.4 represents densities for the subsequent and stationary overflow periods, while Table 7.1 contains distributions of the number of losses during the subsequent and stationary overflow intervals.In both Figure 7.4 and Table 7.1, we may observe rather quick convergence to the limiting distribution.
Example 7.2.In the second example, we are going to use an MMPP fitted to aggregated IP traffic and show how the autocorrelation structure of the traffic influences the overflow period and the probability of consecutive packet losses.For this purpose, a publicly available trace file, recorded at the front range GigaPOP (FRG) aggregation point, which is run by PMA (passive measurement and analysis project, see http://pma.nlanr.net/)has been utilized.Precisely, one million packet headers from the file FRG-1137208198-1.tsh,recorded on Jan 14th, 2006, were used.The mean rate of the traffic is 58.11MB/s, or 71732 packets/s with average packet size of 850 B. Using this traffic sample, the following MMPP parameters were fitted [29]: The average interarrival times are almost the same in the traffic sample and the fitted MMPP (13.940 μs versus 13.941 μs) and, what is important, the MMPP preserves the autocorrelation structure of the original traffic up to three time scales (see Figure 7.5).
We assume that the queue is served at a rate of 80 MB/s (98690 pkts/s), which makes the service (transmission) time to be 10.133 μs and the offered load 73%.We assume also that the buffer can hold 120 packets (100 KB).These settings make the stationary full buffer probability to be 6.55 × 10 −3 , which can be computed by means of the methodology presented in [25].
The transition matrix S b−1 , computed by means of Theorem 4.1, is now 0.60242 0.09548 0.00020 0.29941 0.00247 0.00021 0.97342 0.00054 0.02204 0.00377 0.00260 0.24492 0.07887 0.66704 0.00657 0.00026 0.00677 0.00033 0.98956 0.00305 0.00398 0.13003 0.00014 0.30881 0.55703 and its stationary vector w = (0.00070,0.23479,0.00041,0.75684,0.00723).Now, let us check how the autocorrelation structure of the MMPP influences the duration of the overflow period.For this purpose, let us consider another queue, whose arrival process is simple Poisson and therefore the packet interarrival times are uncorrelated.We set for the same, as previously, arrival rate, service time and buffer size and thus we can compare results.They are shown in Figure 7.7 and Table 7.2.It is evident that in the MMPP case, the probability mass is concentrated around higher values and this causes longer overflow periods.As a consequence, we may observe that in the MMPP queue, the probability of losing a group of consecutive arrivals is much higher than in the Poissonarrival case and this effect is stronger when the size of a group is larger (Table 7.2).For instance, losing 5 packets in row is 12 times more probable in the MMPP queue than in the Poisson one, while losing 10 packets in row is 170 times more probable.

Conclusions and future work
In this paper, the buffer overflow period in a finite-buffer queue fed by a MAP was studied.In particular, formulas for the distribution of the overflow period and the number of arrivals lost during the overflow period were developed.The transient and the stationary cases were both solved.Theoretical results were illustrated via numerical examples.
In addition, it was demonstrated that the autocorrelated structure of the arrival process may have a negative impact on the overflow period and the number of losses that occur consecutively.
As MAP is an analytically tractable model that can capture autocorrelation properties of observed arrival processes and because it is commonly used in network traffic modeling, the results presented herein are likely to be of practical importance.
As regards possible future work on this subject, the first interesting problem is the asymptotic behaviour of the overflow period in a large buffer queue.Based on numerical results, we may expect that as b grows, the duration of the overflow period converges to some limiting distribution.Probably, this limiting distribution has a much simpler form than (3.4), which potentially makes it of practical importance.For Poisson and batch Poisson arrivals, such formulas are already available, see [1,2], respectively.
Now we are going to examine the subsequent and stationary overflow periods assuming b = 50.First of all, we have to compute the transition matrix S b−1 .Using (4.2) gives S b−1 = 0.999202 0.000798 0.509801 0.490199 .(7.3) Andrzej Chydzinski 13

Figure 7 .
Figure 7.6 reports densities of the duration of the first, subsequent, and stationary buffer overflow periods for initial state of the underlying chain 1 and X(0) = b − 1.As in Example 7.1, we can observe rather quick convergence to the limiting distribution.Now, let us check how the autocorrelation structure of the MMPP influences the duration of the overflow period.For this purpose, let us consider another queue, whose arrival process is simple Poisson and therefore the packet interarrival times are uncorrelated.We set for the same, as previously, arrival rate, service time and buffer size and thus we can compare results.They are shown in Figure7.7 and Table7.2.It is evident that in the MMPP case, the probability mass is concentrated around higher values and this causes longer overflow periods.As a consequence, we may observe that in the MMPP queue, the probability of losing a group of consecutive arrivals is much higher than in the Poissonarrival case and this effect is stronger when the size of a group is larger (Table7.2).For

Table 7 .
2. Probability of losing l consecutive packets during the overflow period for MMPP and Poisson arrival systems (both in steady state).Stationary density of the buffer overflow period for MMPP and Poisson arrival systems (Example 7.2).In both systems service times, loads and buffer sizes are the same.