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Using demisubmartingales for the stochastic analysis of networks
Int. J. Electron. Commun. (AEÜ) 69 (2015) 693–698
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International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue
Using demisubmartingales for the stochastic analysis of networks Kishore Angrishi a,∗ , Ulrich Killat b a b
T-Systems International GmbH, 20146 Hamburg, Germany Hamburg University of Technology, 21073 Hamburg, Germany
a r t i c l e
i n f o
Article history: Received 6 August 2014 Accepted 12 December 2014 Keywords: Network calculus End-to-end delay and backlog bounds Doob’s inequality Demimartingales Demisubmartingales
a b s t r a c t Stochastic network calculus is the probabilistic version of the network calculus, which uses envelopes to perform probabilistic analysis of queueing networks. The accuracy of probabilistic end-to-end delay or backlog bounds computed using network calculus has always been a concern. In this article, we propose novel end-to-end probabilistic bounds based on demimartingale inequalities which improve the existing bounds for the tandem networks of GI/GI/1 queues. In particular, we show that reasonably accurate bounds are achieved by comparing the new bounds with the existing results for a network of M/M/1 queues. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Queueing theory is the mathematical study of queues, which generally uses probability mass or density functions to describe arrival traffic and service offered at the network node to compute probabilistic delay or backlog measures. However, with few exceptions, analysis of queueing networks to compute end-to-end probabilistic performance measures is mathematically complex without making simplifying assumptions on arrival traffic or service offered at the network nodes. In most situations however, probabilistic bounds on performance measures are as sufficient as the actual values. Deterministic network calculus is an elegant theory, useful for computing worst-case bounds on end-to-end delay or backlog in queueing networks. Stochastic network calculus is the probabilistic extension of deterministic network calculus, which uses an envelope approach to describe arrival traffic and service offered at the network node. The tightness of the end-to-end probabilistic performance bounds has always been a concern in stochastic network calculus. The concern is mainly due to the use of union bounds for computing the bounds on probabilistic performance measures of the network. Recently, in [8], the author has derived new performance bounds for a GI/GI/1 queue in stochastic network calculus using Doob’s maximal inequality for exponential supermartingales (instead of using union bounds) which are comparable to the exact results of M/M/1 and M/D/1 queues from queueing theory. A general comparison of results for GI/GI/1 queue from
statistical network calculus with the classical queueing theory is made in [15]. More recently in [12,13], authors have used martingales to derive sharp delay bounds for Markov Modulated On-Off arrival process at a single node for FIFO, SP and EDF scheduling schemes. In this paper, we compute end-to-end probabilistic performance bounds for tandem networks of GI/GI/1 queues in stochastic network calculus using demimartingale inequalities [7,18]. The key difference of the approach used in this paper to the work presented in [8] is that we derive performance bounds for a GI/GI/1 queue using effective bandwidth and effective capacity, in contrast to using stochastic processes as in [8]. The relationship between effective bandwidth and stochastic arrival envelope in stochastic network calculus was first explored in [17]. However, the bounds on probabilistic performance measures of the network in [17] are computed using Boole’s inequality and also require a bound on busy period of the scheduler. In [2], we have used effective bandwidth and effective capacity to compute end-to-end probabilitic performance bounds using Boole’s inequality in a more general setting without the requirements on arrival and service processes to have stationary and independent increments. The rest of the paper is structured as follows: In Section 2, we introduce the notion and assumptions used in the paper. In Section 3, we derive the probabilistic end-to-end performance bounds on delay and backlog for the tandem networks of GI/GI/1 queues using effective bandwidth and effective capacity. Brief conclusions are presented in Section 5. 2. Notation and assumptions
∗ Corresponding author. Tel.: +49 1794596542. E-mail address: [email protected] (K. Angrishi). http://dx.doi.org/10.1016/j.aeue.2014.12.008 1434-8411/© 2014 Elsevier GmbH. All rights reserved.
Our time model is discrete with discretization step 0 = 1, i.e., t ∈ N0 = {0, 1, 2, . . .}. We assume that the arrival traffic A and
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K. Angrishi, U. Killat / Int. J. Electron. Commun. (AEÜ) 69 (2015) 693–698
Fig. 1. Network of H concatenated nodes.
the service offered S at a node have stationary and independent increments. In a network of nodes connected in series as shown in Fig. 1, we use non-decreasing, left-continuous processes Ah and Dh to describe the arrivals and the departures at node h, respectively. Ah (s, t) and Dh (s, t) represent the cumulative amount of data seen in an interval (s, t] at input and output of node h, respectively, for any 0 < s < t. For the arrival and departure processes at node h, we assume the initial condition Ah (t, 0) = 0 for t ∈ (− ∞ , 0] and the causal condition Dh (t) ≤ Ah (t), where we denote Ah (0, t) = Ah (t) and Dh (0, t) = Dh (t) for any t > 0. The backlog Bh (t) and delay Wh (t) at time t > 0 in a node h are given by Bh (t) = Ah (t) − Dh (t) and Wh (t) = inf {d ≥ 0 : Ah (t − d) ≤ Dh (t)}, respectively. A stochastic process Sh is said to describe the service offered at node h, if the corresponding arrival and departure processes at node h satisfy for any fixed sample path and t ≥ 0: Ah ⊗ Sh (t) ≤ Dh (t)
(1)
where ⊗ is the min-plus convolution of Ah and Sh which is defined as Ah ⊗ Sh (t) = inf {Ah (0, u) + Sh (u, t)}. Any random process S sat0≤u≤t
isfying the above relationship is referred to as “dynamic F-server” in [6]. The arrival and the service processes are described using statistical envelopes in network calculus. A statistical arrival envelope G for an arrival process A is defined as a non-negative function for all t ≥ 0 satisfying the following condition P{A(t) − G(t) > } ≤ εG ()
(2)
where εG is a non-increasing error function bounding the violation probability of the statistical arrival envelope and ≥ 0. Similarly, a statistical service envelope S describing the service offered at the network node with arrival traffic A and departure traffic D is defined as a non-negative function for all t ≥ 0 satisfying the following condition P{A ⊗ S(t) − D(t) > } ≤ εS ()
(3)
where εS is a non-increasing error function bounding the violation probability of the statistical service envelope. The statistical service envelope from equation (3) is related to the service process from Eq. (1) for all t ≥ 0 by the following expression P{S(t) − S(t) > } ≤ εS ()
1 log E[eA(1) ]
(5)
Similarly, the effective capacity function of a stochastic service process S with stationary and independent increments, for any > 0, is defined as ˇ() = −
1 log E[e−S(1) ]
3. Probabilistic bounds on backlog and delay In this section, we compute probabilistic bounds on backlog and delay in a network of H nodes as shown in Fig. 1 using demimartingale inequalities. Let A1 = A and DH = D be the arrival traffic at the ingress of the network and departure traffic from the egress of the network, respectively. The following theorem provides the probabilistic bounds on end-to-end backlog and delay using the effective bandwidth and effective capacity at each network node h, respectively. Theorem 3.1. Let the service offered at node h in a tandem network be characterized by the stochastic service process Sh with the corresponding effective capacity function ˇh (), for h = 1, . . ., H. Let A be the arrival process with effective bandwidth ˛() and D be the departure process from the tandem network with H nodes. Then we have the following bounds. 1. Backlog bound: The probabilistic bound on the backlog in a network, for any t > 0, is given by
P B(t) > x
(6)
Then the statistical arrival and service envelopes in terms of effective bandwidth of the probabilistic arrival process [17] and effective capacity of the service process observed at a network node [3] are given as G(t) = ˛()t and S(t) = ˇ()t, respectively; they satisfy the appropriate conditions in Eqs. (2) and (4) with
≤ ε˜ (x)
(7)
2. Delay bound: The probabilistic bound on the delay in a network, for any t > 0, is given by
P W (t) > d
≤ ε˜ (˛( ∗ )d)
(8)
where ε˜ is an error function, for any x ≥ 0, given as:
(4)
In this paper, we use the notion of effective bandwidth ˛() [16] and effective capacity ˇ() [20,22,3] from large deviations theory to describe the stochastic arrival traffic and the service offered at a node, respectively. The effective bandwidth of an arrival traffic A with stationary and independent increments from [16], for any > 0, is given as ˛() =
the error function ε() = e− . The main advantage of using network calculus to do performance analysis of networks is that the network calculus allows to model a network of nodes as a single virtual node. The stochastic network service process Snet characterizing the service offered in a single virtual network node, which represents a network of H nodes connected in series as shown in Fig. 1, can be computed for any fixed sample path using the min-plus convolution of the stochastic service process Sh of constituting nodes for h = 1, . . ., H, i.e., Snet = S1 ⊗ S2 ⊗ · · · ⊗ SH [6,11]. The corresponding statistical network service envelope is given as Snet = S1 ⊗ S2 ⊗ · · · ⊗ SH , where Sh is the statistical service envelope describing the service offered at node h, for h = 1, 2, . . ., H. We assume that the arrival traffic A1 at the ingress of the network and the stochastic service processes Sh , for h = 1, . . ., H, characterizing the service offered at the nodes of the network are independent of each other.
ε˜ (x) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
e−
∗x
if H = 1
( ∗ x − (H − 1))h H
e−(
∗ x−(H−1))
h!
if H > 1
h=0
and x ≥
(9) H−1 ∗
and * defined by min h∗ where h∗ = { : ˇh () = ˛(), > 0} 1≤h≤H
The proof of the theorem relies on applying demimartingale inequalities to compute probabilistic bounds. The key observation is that certain functions of the random arrival and service processes together with effective capacity and effective bandwidth, respectively, form demisubmartingales 1 at = * . This is shown using the following lemma.
1
A sequence
Sn , n ≥ 1
is said to be a demisubmartingale if E[(Sj+1 − Sj )f(S1 , . . .,
Sj )] ≥ 0, j = 1, 2, . . . for every nonnegative coordinatewise nondecreasing function f whenever the expectation is defined.
K. Angrishi, U. Killat / Int. J. Electron. Commun. (AEÜ) 69 (2015) 693–698
Lemma 3.2. Let A be the arrival traffic with effective bandwidth ˛() at a network node offering a stochastic service characterized by a service process S with effective capacity ˇ(). If the arrival and the service processes have stationary and independent increments, then the random processes X(, t) = eA(,t)−˛()(t−) , Y(, t) = e˛()(t−)−S(,t) , Y ∗ (t) = sup Y (v, t) and Y1/ (, t) are demisub-
the expectation is defined E[(Z( − 1, t) − Z(, t))f(Z(t − 1, t), Z(t − 2, t), . . ., Z(, t))] ≥ 0 for t ∈ N = {1, 2, . . .}. E[(Z( − 1, t) − Z(, t))f (Z(t − 1, t), . . ., Z(, t))] =
martingales with respect to t at = * and Z(, t) = eA(,t−d)−S(,t) is a demisubmartingale with respect to at = * for 0 ≤ ≤ t where t ∈ N = {1, 2, . . .}, d ≥ 0, > 1, > 0 and * = { : ˇ() = ˛()}.
E
1 1 Y (, t + 1) − Y (, t)
=
E
1 Y (, t)f
=
⎛ =
= ≥
⎝e
e
˛() − ˇ
˜ ˜ ˇ ˜ (˜ ) ˛( )−
−1 E
g
=E
E
0≤v≤t+1
≥
= =
0≤v≤t
sup Y (v, t)Y (t, t + 1)
E E
sup e{˛()(t−v)−S(v,t)}+{˛()−S(t,t+1)} − sup Y (v, t)
E
0≤v≤t
=
f (Y (1), . . ., Y (t))
0≤v≤t
0≤v≤t
e˛()−S(t,t+1) − 1
− sup Y (v, t)
f (Y ∗ (1), . . ., Y ∗ (t))
f (Y ∗ (1), . . ., Y ∗ (t))
0≤v≤t
E [Y ∗ (t)f (Y ∗ (1), . . ., Y ∗ (t))]
e˛()−ˇ() − 1 E [Y ∗ (t)f (Y ∗ (1), . . ., Y ∗ (t))] = 0 at = ∗
The inequality at the third step is due to the reduction of the upper bound of the supremum from t + 1 to t. The last equality is from the definition of * . This proves that Y* (t) is a demisubmartingale. To prove that Z(, t) is demisubmartingale with respect to at = * for > 0, 0 ≤ ≤ t, we need to show that for every coordinatewise non-decreasing, non-negative function f whenever
∗
sup Y (v, u) > e 0≤v≤u≤t
∗
∗
=P
0≤v≤u − ∗
sup (Y (u)) > e 0≤u≤t
− ∗
=E
sup
1 Y (v, t)
e−
∗
0≤v≤t
(11)
∗
(by Doob s maximal inequality)
sup Y (v, t) e
sup 0≤v≤u≤t
∗
=P
1
Y (t)
E
(12)
e−
∗
¨ (by applying Holder s inequality)
The first inequality is due to Doob’s inequality for demisubmartingales (cf. Corollary 2.4 in [7]). Application of Hölder’s inequality to Y1/ (cf. Theorem 3.6. especially equation (3.10) in [18]) provides the second inequality. From Eq. (12), the probability of interest is also bounded by the following limit.
P
∗
>e
0≤v≤t
sup Y (v, u)
−1
>
sup e{˛()(t+1−v)−S(v,t+1)} − sup Y (v, t)
≤ E [Y ∗ (t)] e
f (Y ∗ (1), . . ., Y ∗ (t))
0≤v≤t
(10)
sup
0≤v≤t+1
1 E [St ] x
˛( ∗ )(u − v) − S(v, u)
0≤u≤t
≤
=P
E[(Y ∗ (t + 1) − Y ∗ (t))f (Y ∗ (1), . . ., Y ∗ (t))]
=
sup
for ˜ =
≤
sup Sk > x
0≤v≤u≤t
sup Y (v, t + 1) − sup Y (v, t)
⎞
1 1 − 1⎠ E g Y (, + 1), . . ., Y (, t)
e˛()−ˇ() − 1 E [Z(, t)f (Z(t − 1, t), . . ., Z(, t))] = 0 at = ∗
P
0 at ˜ = ∗
E
For S(t), ˛( * ) and any > 0, we then get
Second and third equalities are due to the fact that the process Y(t) has stationary and independent increments and E[e−S(t,t+1) ] = e−ˇ() (cf., Eq. (6)), respectively. The final inequality ˜ = ˛(), ˜ ˜ > 0} and from the is due to the definition of ∗ = { : ˇ() ˜ ≥ ˛() ˜ for > 1. property of effective bandwidth cf., [16], i.e., ˛( ) This proves that Y1/ (, t) is a demisubmartingale.
=
E eA(−1,) E e−S(−1,) − 1 E [Z(, t)f (Z(t − 1, t), . . ., Z(, t))]
1 1 Y (, + 1), . . ., Y (, t)
1 1 Y (, + 1), . . ., Y (, t)
In the sequel we will make use ofDoob’s maximal inequality for demisubmartigales Sk , k = 1, . . ., t [7]
Second and third equalities are due to the fact that the process Z(t) has stationary and independent increments and E[e−S(−1,) ] = e−ˇ() , E[eA(−1,) ] = e˛() (cf., Eqs. (6), (5)), respectively. The final equality is due to the definition of * . This proves that Z(, t) is a demisubmartingale with respect to .
1 1 Y (, + 1), . . ., Y (, t)
=
eA(−1,)−S(−1,) − 1 Z(, t)f (Z(t − 1, t), . . ., Z(, t))
1≤k≤t
1 ˛() − S(t, t + 1)) − 1 E Y (, t)f E e
eA(−1,t−d)−S(−1,t) − eA(,t−d)−S(,t) f (Z(t − 1, t), . . ., Z(, t))
=
P
1 1 Y (, + 1), . . ., Y (, t)
f
1 Y (t, t + 1) − 1
E
= E
0≤v≤t
Proof. To prove that X(, t), Y(, t), Y1/ (, t) and Y* (t) are demisubmartingales with respect to t ∈ N = {1, 2, . . .} at = * for > 0, 0 ≤ ≤ t and any > 1, we need to show that for every coordinatewise non-decreasing, non-negative function f whenever the expectation is defined E[(X(, t + 1) − X(, t))f(X(, + 1), X(, + 2), . . ., X(, t))] ≥ 0, corresponding statements for Y(, t), Y1/ (, t) and Y* (t) will hold for t ∈ N = {1, 2, . . .}, 0 ≤ ≤ t and any > 1. As the proof for X(, t) and Y(, t) follow the same lines as Y1/ (, t) for = 1, we will provide the proofs only for Y1/ (, t) and Y* (t).
695
˛( ∗ )(u − v) − S(v, u)
≤ eE [Y (t)] e− = ee−
∗
∗
, as → ∞,
>
−1
, as E [Y (t)] = 1 at = ∗
→e (13)
The proof of Theorem 3.1 also relies on Lemma 4.1 from [14], which states that for any two non-negative independent random variables F and G with P(F > ) ≤ f() and P(G > ) ≤ g() where f() and g() are non-negative, decreasing functions for any ≥ 0, then
P F +G>
f˜ ( − u)dg˜ (u)
≤1−
(14)
0 −
where f˜ () = 1 − [f ()] , g˜ () = 1 − [g()]− and [a]− = min(1, a) for any a ≥ 0. Proof of Theorem 3.1 : We now provide the proof for the probabilistic end-to-end delay bound. For single hop (H = 1), the proof
696
K. Angrishi, U. Killat / Int. J. Electron. Commun. (AEÜ) 69 (2015) 693–698
is straightforward and can be shown for a fixed sample path, t ≥ 0 and > 0 as follows:
P
=P
W (t) > d
= P ≤
sup
e
A(t − d) − D(t) > 0
∗ A(u,t−d)− ∗ S (u,t) 1
0≤u≤t
E e
∗ {A(t−d)−S (t)} 1
= e
≤P
time. However, the results can also be used in continuous time: Lemma 3.2 can be proven with any discretization step 0 and therefore is not limited to discretization step 0 = 1. Furthermore, as indicated in [19] Chapter 5.5, Doob’s maximal inequality and the use of Hölder’s inequality in the derivation of Eq. (13) also applicable for continuous time demisubmartingales and hence our result of Eq. (16) also applies to time-continuous problems. To analyse the accuracy of the new probabilistic end-to-end delay bound from Theorem 3.1, we compare it with the existing probabilistic bounds from network calculus and results from queueing theory for a network of M/M/1 queues. The customers arrive at rate and the server works at rate . We denote the utilization factor by = /, and assume for stability that < 1. The effective bandwidth and effective capacity of the arrival and service processes (Poisson process) are (e − 1)/ and (1 − e− )/, respectively, with = − log . We consider a special case of the network from Fig. 1 with H M/M/1
A(t − d) − A ⊗ S1 (t) > 0
(15)
>1
∗ ˛( ∗ )(t−d)− ∗ ˇ( ∗ )t
= e−
∗ ˛( ∗ )d
The first inequality is from the definition of stochastic network service process from Eq. (1). The final inequality is due to Doob’s inequality for demisubmartingale (Z(, t) with respect to ) from Eq. (10). The last two steps are due to our assumption about arrival A(t) and service S(t) processes having independent increments and from the definition of * , respectively. One can also arrive at the same result for H = 1 using a node operating point approach as described in [3]. In a multi-hop network (H > 1), for a fixed sample path, t ≥ 0 and > 0, we have, P
W (t) > d
= P
=P
A(t − d) − D(t) > 0
A(t − d) − A ⊗ Snet (t) > 0
A(t − d) − A ⊗ S1 ⊗ · · · ⊗ SH (t) > 0
= P
{A(k1 , t − d) −
sup
H−1
0≤k1 ≤k2 ≤···≤kH ≤t
≤
≤P
P
˛(h∗ )(kh+1
− kh ) −
{A(k1 , t − d) − ˛( )(t − d − k1 ) +
0≤k1 ≤k2 ≤···≤kH ≤t
≤
P
sup
≤
e
˛(H∗ )(t
− kH ) − SH (kH , t) +
A(k1 , t − d) − ˛( ∗ )(t − d − k1 )
+ sup 0≤kH ≤t
H h ( ∗ ˛( ∗ )d − (H − 1))
h!
The first inequality at the second step is from the definition of stochastic network service process from Eq. (1). The inequality at the fifth step is obtained by choosing ∗ = min h∗ ; note that h∗ 1≤h≤H
is the value of h at node h for which the effective bandwidth ˛ and the effective capacity ˇh satisfy the condition ˛(h∗ ) = ˇh (h∗ ). The justification for this equation lies in the (approximate) invariance of the effective bandwidth ˛() [4]. The inequality at the sixth step is from a property of supremum operation2 [14]. We get the final inequality from Lemma 3.2, (11), (13), (14) and by observing ∗ ∗ − ∗ ˛( ∗ ) that e h h ≤ e− ˛( ) holds for 1 ≤ h ≤ H. The proof is obtained by a complete induction over H. The final expression is valid only for ˛( * )d ≥ (H − 1)/ * . This constraint is a consequence of the [a]− operation in Eq. (14). The proof for the probabilistic bound on endto-end backlog is its immediate variation. It can be observed that setting H = 1 in Eq. (16), one gets the ∗ ∗ probability bound to be (1 + ∗ ˛( ∗ )d)e− ˛( )d . Though the bound is valid, it can be easily verified that it is worse than the bound ∗ ∗ e− ˛( )d from Eq. (15). This discrepancy between the two bounds is due to the fact that for the bound from Eq. (15) the stochastic process sup {A(u, t − d) − S1 (u, t)} is directly used to determine 0≤u≤t−d
the delay bound, in contrast to the bound from Eq. (16) where the arrival process with effective bandwidth and service process with effective capacity are used individually to establish the result. 4. Experiments using the new delay bound Complying with most of the relevant literature in this field we have based our derivation using demisubmartingales on discrete
sup {X(s) + Y (s)} ≤ sup {X(s)} + sup {Y (s)}. 0≤s≤t
0≤s≤t
H−1
˛(h∗ )(kh+1
˛(h∗ )(kh+1
− kh ) − Sh (kh , kh+1 ) } > 0
h=1
0≤s≤t
H−1
˛(H∗ )(t − kH ) − SH (kH , t)
h=0
2
− kH ) − SH (kH , t) +
H−1
(16)
∗
− kh ) − Sh (kh , kh+1 ) } > ˛( )d
h=1
0≤k1 ≤t
−( ∗ ˛( ∗ )d−(H−1))
− kH ) +
˛(H∗ )(t
h=1 ∗
sup
˛(H∗ )(t
+
h=1
sup
0≤kh ≤kh+1 ≤t
˛(h∗ )(kh+1 − kh ) − Sh (kh , kh+1 )
> ˛( ∗ )d
queues connected in series to analyse the accuracy of our endto-end network calculus delay bound. A Poisson flow with rate traverses through the entire network. Let each queue in the network be served by a service process S with effective capacity ˇ() and let the service processes at all nodes of the network be independent of each other. It is known from queueing theory that the exact distribution of steady state end-to-end delay of the through flow W(t) in a M/M/1 queueing network is given by
P W (t) > d
=
H−1 h ((1 − )d)
h!
e−(1− )d
(17)
h=0
The equation is obtained from an H-fold convolution of the (exponential) probability density function of delay for a single M/M/1 node, followed by an integration in the limits from d to infinity. The best available end-to-end delay bound W(t) from network calculus [8] in discrete time domain with discretization step 0 = 1 for the through flow in a network of tandem queues with both the arrival and the service processes having stationary and independent increments is given by
P W (t) > d
= inf
0≤≤ ∗
1 1 − e−{ˇ()−˛()}
H
e−˛()d
(18)
Eq. (17) has been derived under steady state conditions. Eqs. (16) and (18) hold for all values of t. As this includes very large values of t the bounds also address steady state situations. In Fig. 2, we illustrate the violation probability of the delay bounds (d = 4.5 ms and = 25 kbps) from Eqs. (8) and (9) as curve (b) along with the exact results from Eq. (17) as curve (a) and existing delay bounds from stochastic network calculus using moment generating functions [11,8] from Eq. (18) as curve (c) for a fixed utilization factor = 0.7 at each of the H queues. It can be observed that the new results provide better bounds than the existing bounds and also follow the shape of the exact results from queueing theory.
K. Angrishi, U. Killat / Int. J. Electron. Commun. (AEÜ) 69 (2015) 693–698
The above condition can be expressed in terms of effective bandwidth function for any t ≥ 0 and > 0, as shown below:
0
10
Violation Probability
1 (t − s)
∞
P(eA(s,t) > z)dz ≤ 0
−5
1 (t − s)
∞
P(eG(s,t) > z)dz 0
(20)
˛A (, t − s) ≤ ˛G (, t − s)
10
Similarly, we need a random process B with stationary and independent increments to stochastically lower bound the actual random service process S, i.e., B(s, t) ≤ st S(s, t), to enable us to employ the new probabilistic end-to-end delay bound from Theorem 3.1 for the connection admission control. The condition from Eq. (19) can be expressed in terms of effective capacity function for all t ≥ s ≥ 0 and > 0, as shown below:
−10
10
a) Exact result b) Bound using Theorem 3.1 c) Existing bound from network calculus −15
10
697
1
2
3
4
5
6
7 8 9 10 11 12 13 14 15 16 Number of Hops (H)
1 − (t − s)
∞
P(e
B(s,t)
0
P(A(s, t) > z) ≤ P(G(s, t) > z)
(19)
∞
P(eS(s,t) > z)dz 0
(21) Let the through and the cross arrival flows be modelled as an aggregate of Markov modulated On-Off arrival processes. This type of process has been used to model VoIP [1]. Markov modulated onoff process can be in “On” state or “Off” state for a random time interval which is negative exponentially distributed with average E[Ton ] and E[Toff ], respectively. In “On” state, arrival traffic transmits data at a constant rate R and no data is transmitted in “Off” state. This process is stochastically upper bounded by a process with stationary and independent increments which results from the original process by taking the time independent version of effective bandwidth ˛() = lim ˛(, t). Thus this derived process t→∞
can take the role of G in Eq. (20). ˛() for any > 0 is given by [9] ˛() =
1 (R − r10 − r01 + 2
2
(R − r10 + r01 ) + 4r10 r01 )
where r10 = 1/E[Ton ] and r01 = 1/E[Toff ]. Similarly, the leftover service process S at each network node h determined by general blind multiplexing model [11] is given by S(t) = Ct − AM (t) and there exists a random process B with stationary and independent increments, whose effective capacity is ˇh () = C − M˛() which stochastically lower bounds the leftover service process S at each network node. Now that we have two random processes with stationary and independent increments to describe the arrival and the service processes, respectively, we can use Eq. (8) in Theorem 3.1 and Eq. (18)
320 Bound using Theorem 3.1 Existing bound from network calculus Simulation results
300
Number through flows (N)
To illustrate the practical application of the new probabilistic end-to-end delay bound from Theorem 3.1, we employ it to perform connection admission control in a data network as shown in Fig. 3. An aggregate of N independent through arrival flows AN traverses H nodes connected in tandem and at each network node the through arrival flows share the resources with an aggregate of M independent cross arrival flows AM . Each network node is assumed to have a FIFO queue of infinite size and is served at a constant service rate C. The network is assumed to be stable, i.e., the capacity C is greater than the average arrival rate at each node. The simplified network in Fig. 3 used for the connection admission control can be viewed as segment of a possibly larger feed-forward network. There are some algorithms to turn non feed-forward networks to Feed-Forward Networks, namely, spanning tree and turn prohibition algorithm [21,10]. However, two critical assumptions are made regarding the flow path, namely, (i) arrival flows follow the same path for the entire duration of their transmission, (ii) effective bandwidth description of the cross flows at each network node along the path of through flow is known. The goal is to perform connection admission control of the through arrival flows AN in the tandem data network shown in Fig. 3 for a given probabilitic end-to-end delay bound with a fixed number M of cross arrival flows AM at each network node for different number of hops H in the tandem network. The main issue in the application of the new probabilistic end-to-end delay bound from Theorem 3.1 is its restriction to the networks with arrival and service processes having stationary and independent increments. However, if there exists a random process G with stationary and independent increments which stochastically upper bounds the actual random arrival process A, we can employ the random process G with stationary and independent increments for the analysis instead of the random arrival process A. A random arrival process A is said to be stochastically upper bounded by another random process G, and we write this as A(s, t) ≤ st G(s, t), if for all real z and all t ≥ s ≥ 0 the following condition is satisfied.
ˇB (, t − s) ≤ ˇS (, t − s)
Fig. 2. Violation probability of the end-to-end delay bound (d = 4.5 ms and = 25 kbps) in an M/M/1 network as a function of the number of nodes; utilization factor at each node is = 0.7.
Fig. 3. Network of H concatenated nodes with cross and through arrival flows.
1 > z)dz ≤ − (t − s)
280
260
240
220
200
180
2
4
6
8
10
12
14
16
Number of hops (H) Fig. 4. Maximum number of through flows allowable in the network for end-to-end delay bound of 300 ms with violation probability of 10−6 as a function of the number of nodes.
698
K. Angrishi, U. Killat / Int. J. Electron. Commun. (AEÜ) 69 (2015) 693–698
to perform connection admission control in a network as shown in Fig. 3. In the example, we determine the maximum number of through flows N allowable in the network without violating the probabilistic end-to-end delay bound of 300 ms with the violation probability of 10−6 . The capacity of the server C at each hop is set to 10 Mbps and the number of cross flows M at each hop set to a constant of 100 flows. To model a voice source using Markov modulated On-Off flows we set the parameters for each flow E[Toff ] = 650 ms, E[Ton ] = 352 ms and peak rate R = 64 kbps. Each Markov modulated On-Off flow produces data at an average rate m = 22.48 kbps. Fig. 4 shows the maximum number of through flows N allowable in the network without violating the probabilistic end-to-end delay bound of 300 ms with the violation probability of 10−6 as a function of increasing number of hops H. It can be seen that the new probabilistic delay bounds from Eqs. (8) and (9) provide better results than the existing network calculus result on probabilistic delay bound from Eq. (18). The new delay bounds benefit from the property of a sum of independent random variables defined in Eq. (14) and the demimartingales bounds, whereas the existing results from Eq. (18) provide loose delay bounds due to the use of union bounds. The simulation results give an upper bound for N determined from the first observation of a violation of the given delay bound of 300 ms 5. Conclusions In this paper we used demimartingale inequalities to compute end-to-end probabilistic delay and backlog bounds within the framework of network calculus. The tightness of the computed end-to-end probabilistic performance bounds is explored by comparing new bounds with the exact results from queueing theory for a network of M/M/1 queues. References [1] International Telecommunication Union. Recommendation ITU-T P.59, Artificial Conversational Speech; 1993. [2] Angrishi K. An end-to-end stochastic network calculus with effective bandwidth and effective capacity. Comput Netw 2013;57(1):78–84.
[3] Angrishi K, Killat U. Analysis of a real-time network using statistical network calculus with effective bandwidth and effective capacity. In: Proceedings of 14. GI/ITG Konferenz Messung, Modellierung und Bewertung von Rechen- und Kommunikationssystemen (MMB 2008). 2008. p. 1–15. [4] Angrishi K, Killat U. On the threshold for observing approximate invariance of effective bandwidth. In: Proceedings of 2008 international symposium on performance evaluation of computer and telecommunication systems (SPECTS 2008). 2008. p. 511–8. [3] Angrishi K, Killat U. A node operating point approach for stochastic analysis with network calculus. J Netw 2012;7:1739–48. [6] Chang C-S. Performance guarantees in communication networks. SpringerVerlag; 2000. [7] Christofides T. Maximal inequalities for demimartingales and a strong law of large numbers. Stat Probab Lett 2000;50:357–63. [8] Ciucu F. Network calculus delay bounds in queueing networks with exact solutions. In: Proceedings of international teletraffic congress (ITC-20). 2007. p. 495–506. [9] Courcoubetis C, Weber R. Buffer overflow asymptotics for a switch handling many traffic sources. J Appl Probab 1996;33:886–903. [10] Echague J, Villadangos J, Cholvi V, Prieto M. Transforming general networks into feed-forward by using turn-prohibition. Comput Commun 2008;31(9):1824–31. [11] Fidler M. An end-to-end probabilistic network calculus with moment generating functions. In: Proceedings of IWQoS. 2006. p. 261–70. [12] Poloczek F, Ciucu F. Scheduling analysis with martingales. In: Proceedings of IFIP performance. 2014. p. 56–72. [13] Ciucu F, Poloczek F, Schmitt J. Sharp per-flow delay bounds for bursty arrivals: the case of FIFO, SP, and EDF scheduling. In: Proceedings of IEEE infocom. 2014. p. 1896–904. [14] Jiang Y. A basic stochastic network calculus. In: Proceedings of ACM SIGCOMM. 2006. p. 123–34. [15] Jiang Y. Network calculus and queueing theory: two sides of one coin. In: Proceedings of the fourth international ICST conference on performance evaluation methodologies and tools. 2009. p. 37:1–12. [16] Kelly FP. Notes on effective bandwidths. In: Stoch Netw Theory Appl. Oxford University Press; 1996. p. 141–68. [17] Li C, Burchard A, Liebeherr J. A network calculus with effective bandwidth. IEEE/ACM Trans Netw 2007;15(6):1442–53. [18] Prakasa Rao BLS. On some maximal inequalities for demisubmartingales and n-demisupermartingales. J Inequal Pure Appl Math 2007;8(4):1–37. [19] Prakasa Rao BLS. Associated sequences, demimartingales and nonparametric inference. In: Probability and its applications. Birkhauser Verlag GmbH; 2012. [20] Shakkottai S, Kumar A, Karnik A, Anvekar A. TCP performance over endto-end rate control and stochastic available capacity. IEEE/ACM Trans Netw 2001;9(4):377–91. [21] Starobinski D, Karpovsky M, Zakrevski L. Application of network calculus to general topologies using turn-prohibition. IEEE/ACM Trans Netw 2003;11:411–21. [22] Wu D, Negi R. Effective capacity: a wireless link model for support of quality of service. IEEE Trans Wirel Commun 2003;2(4):630–43.
Contents lists available at ScienceDirect
International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue
Using demisubmartingales for the stochastic analysis of networks Kishore Angrishi a,∗ , Ulrich Killat b a b
T-Systems International GmbH, 20146 Hamburg, Germany Hamburg University of Technology, 21073 Hamburg, Germany
a r t i c l e
i n f o
Article history: Received 6 August 2014 Accepted 12 December 2014 Keywords: Network calculus End-to-end delay and backlog bounds Doob’s inequality Demimartingales Demisubmartingales
a b s t r a c t Stochastic network calculus is the probabilistic version of the network calculus, which uses envelopes to perform probabilistic analysis of queueing networks. The accuracy of probabilistic end-to-end delay or backlog bounds computed using network calculus has always been a concern. In this article, we propose novel end-to-end probabilistic bounds based on demimartingale inequalities which improve the existing bounds for the tandem networks of GI/GI/1 queues. In particular, we show that reasonably accurate bounds are achieved by comparing the new bounds with the existing results for a network of M/M/1 queues. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Queueing theory is the mathematical study of queues, which generally uses probability mass or density functions to describe arrival traffic and service offered at the network node to compute probabilistic delay or backlog measures. However, with few exceptions, analysis of queueing networks to compute end-to-end probabilistic performance measures is mathematically complex without making simplifying assumptions on arrival traffic or service offered at the network nodes. In most situations however, probabilistic bounds on performance measures are as sufficient as the actual values. Deterministic network calculus is an elegant theory, useful for computing worst-case bounds on end-to-end delay or backlog in queueing networks. Stochastic network calculus is the probabilistic extension of deterministic network calculus, which uses an envelope approach to describe arrival traffic and service offered at the network node. The tightness of the end-to-end probabilistic performance bounds has always been a concern in stochastic network calculus. The concern is mainly due to the use of union bounds for computing the bounds on probabilistic performance measures of the network. Recently, in [8], the author has derived new performance bounds for a GI/GI/1 queue in stochastic network calculus using Doob’s maximal inequality for exponential supermartingales (instead of using union bounds) which are comparable to the exact results of M/M/1 and M/D/1 queues from queueing theory. A general comparison of results for GI/GI/1 queue from
statistical network calculus with the classical queueing theory is made in [15]. More recently in [12,13], authors have used martingales to derive sharp delay bounds for Markov Modulated On-Off arrival process at a single node for FIFO, SP and EDF scheduling schemes. In this paper, we compute end-to-end probabilistic performance bounds for tandem networks of GI/GI/1 queues in stochastic network calculus using demimartingale inequalities [7,18]. The key difference of the approach used in this paper to the work presented in [8] is that we derive performance bounds for a GI/GI/1 queue using effective bandwidth and effective capacity, in contrast to using stochastic processes as in [8]. The relationship between effective bandwidth and stochastic arrival envelope in stochastic network calculus was first explored in [17]. However, the bounds on probabilistic performance measures of the network in [17] are computed using Boole’s inequality and also require a bound on busy period of the scheduler. In [2], we have used effective bandwidth and effective capacity to compute end-to-end probabilitic performance bounds using Boole’s inequality in a more general setting without the requirements on arrival and service processes to have stationary and independent increments. The rest of the paper is structured as follows: In Section 2, we introduce the notion and assumptions used in the paper. In Section 3, we derive the probabilistic end-to-end performance bounds on delay and backlog for the tandem networks of GI/GI/1 queues using effective bandwidth and effective capacity. Brief conclusions are presented in Section 5. 2. Notation and assumptions
∗ Corresponding author. Tel.: +49 1794596542. E-mail address: [email protected] (K. Angrishi). http://dx.doi.org/10.1016/j.aeue.2014.12.008 1434-8411/© 2014 Elsevier GmbH. All rights reserved.
Our time model is discrete with discretization step 0 = 1, i.e., t ∈ N0 = {0, 1, 2, . . .}. We assume that the arrival traffic A and
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Fig. 1. Network of H concatenated nodes.
the service offered S at a node have stationary and independent increments. In a network of nodes connected in series as shown in Fig. 1, we use non-decreasing, left-continuous processes Ah and Dh to describe the arrivals and the departures at node h, respectively. Ah (s, t) and Dh (s, t) represent the cumulative amount of data seen in an interval (s, t] at input and output of node h, respectively, for any 0 < s < t. For the arrival and departure processes at node h, we assume the initial condition Ah (t, 0) = 0 for t ∈ (− ∞ , 0] and the causal condition Dh (t) ≤ Ah (t), where we denote Ah (0, t) = Ah (t) and Dh (0, t) = Dh (t) for any t > 0. The backlog Bh (t) and delay Wh (t) at time t > 0 in a node h are given by Bh (t) = Ah (t) − Dh (t) and Wh (t) = inf {d ≥ 0 : Ah (t − d) ≤ Dh (t)}, respectively. A stochastic process Sh is said to describe the service offered at node h, if the corresponding arrival and departure processes at node h satisfy for any fixed sample path and t ≥ 0: Ah ⊗ Sh (t) ≤ Dh (t)
(1)
where ⊗ is the min-plus convolution of Ah and Sh which is defined as Ah ⊗ Sh (t) = inf {Ah (0, u) + Sh (u, t)}. Any random process S sat0≤u≤t
isfying the above relationship is referred to as “dynamic F-server” in [6]. The arrival and the service processes are described using statistical envelopes in network calculus. A statistical arrival envelope G for an arrival process A is defined as a non-negative function for all t ≥ 0 satisfying the following condition P{A(t) − G(t) > } ≤ εG ()
(2)
where εG is a non-increasing error function bounding the violation probability of the statistical arrival envelope and ≥ 0. Similarly, a statistical service envelope S describing the service offered at the network node with arrival traffic A and departure traffic D is defined as a non-negative function for all t ≥ 0 satisfying the following condition P{A ⊗ S(t) − D(t) > } ≤ εS ()
(3)
where εS is a non-increasing error function bounding the violation probability of the statistical service envelope. The statistical service envelope from equation (3) is related to the service process from Eq. (1) for all t ≥ 0 by the following expression P{S(t) − S(t) > } ≤ εS ()
1 log E[eA(1) ]
(5)
Similarly, the effective capacity function of a stochastic service process S with stationary and independent increments, for any > 0, is defined as ˇ() = −
1 log E[e−S(1) ]
3. Probabilistic bounds on backlog and delay In this section, we compute probabilistic bounds on backlog and delay in a network of H nodes as shown in Fig. 1 using demimartingale inequalities. Let A1 = A and DH = D be the arrival traffic at the ingress of the network and departure traffic from the egress of the network, respectively. The following theorem provides the probabilistic bounds on end-to-end backlog and delay using the effective bandwidth and effective capacity at each network node h, respectively. Theorem 3.1. Let the service offered at node h in a tandem network be characterized by the stochastic service process Sh with the corresponding effective capacity function ˇh (), for h = 1, . . ., H. Let A be the arrival process with effective bandwidth ˛() and D be the departure process from the tandem network with H nodes. Then we have the following bounds. 1. Backlog bound: The probabilistic bound on the backlog in a network, for any t > 0, is given by
P B(t) > x
(6)
Then the statistical arrival and service envelopes in terms of effective bandwidth of the probabilistic arrival process [17] and effective capacity of the service process observed at a network node [3] are given as G(t) = ˛()t and S(t) = ˇ()t, respectively; they satisfy the appropriate conditions in Eqs. (2) and (4) with
≤ ε˜ (x)
(7)
2. Delay bound: The probabilistic bound on the delay in a network, for any t > 0, is given by
P W (t) > d
≤ ε˜ (˛( ∗ )d)
(8)
where ε˜ is an error function, for any x ≥ 0, given as:
(4)
In this paper, we use the notion of effective bandwidth ˛() [16] and effective capacity ˇ() [20,22,3] from large deviations theory to describe the stochastic arrival traffic and the service offered at a node, respectively. The effective bandwidth of an arrival traffic A with stationary and independent increments from [16], for any > 0, is given as ˛() =
the error function ε() = e− . The main advantage of using network calculus to do performance analysis of networks is that the network calculus allows to model a network of nodes as a single virtual node. The stochastic network service process Snet characterizing the service offered in a single virtual network node, which represents a network of H nodes connected in series as shown in Fig. 1, can be computed for any fixed sample path using the min-plus convolution of the stochastic service process Sh of constituting nodes for h = 1, . . ., H, i.e., Snet = S1 ⊗ S2 ⊗ · · · ⊗ SH [6,11]. The corresponding statistical network service envelope is given as Snet = S1 ⊗ S2 ⊗ · · · ⊗ SH , where Sh is the statistical service envelope describing the service offered at node h, for h = 1, 2, . . ., H. We assume that the arrival traffic A1 at the ingress of the network and the stochastic service processes Sh , for h = 1, . . ., H, characterizing the service offered at the nodes of the network are independent of each other.
ε˜ (x) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
e−
∗x
if H = 1
( ∗ x − (H − 1))h H
e−(
∗ x−(H−1))
h!
if H > 1
h=0
and x ≥
(9) H−1 ∗
and * defined by min h∗ where h∗ = { : ˇh () = ˛(), > 0} 1≤h≤H
The proof of the theorem relies on applying demimartingale inequalities to compute probabilistic bounds. The key observation is that certain functions of the random arrival and service processes together with effective capacity and effective bandwidth, respectively, form demisubmartingales 1 at = * . This is shown using the following lemma.
1
A sequence
Sn , n ≥ 1
is said to be a demisubmartingale if E[(Sj+1 − Sj )f(S1 , . . .,
Sj )] ≥ 0, j = 1, 2, . . . for every nonnegative coordinatewise nondecreasing function f whenever the expectation is defined.
K. Angrishi, U. Killat / Int. J. Electron. Commun. (AEÜ) 69 (2015) 693–698
Lemma 3.2. Let A be the arrival traffic with effective bandwidth ˛() at a network node offering a stochastic service characterized by a service process S with effective capacity ˇ(). If the arrival and the service processes have stationary and independent increments, then the random processes X(, t) = eA(,t)−˛()(t−) , Y(, t) = e˛()(t−)−S(,t) , Y ∗ (t) = sup Y (v, t) and Y1/ (, t) are demisub-
the expectation is defined E[(Z( − 1, t) − Z(, t))f(Z(t − 1, t), Z(t − 2, t), . . ., Z(, t))] ≥ 0 for t ∈ N = {1, 2, . . .}. E[(Z( − 1, t) − Z(, t))f (Z(t − 1, t), . . ., Z(, t))] =
martingales with respect to t at = * and Z(, t) = eA(,t−d)−S(,t) is a demisubmartingale with respect to at = * for 0 ≤ ≤ t where t ∈ N = {1, 2, . . .}, d ≥ 0, > 1, > 0 and * = { : ˇ() = ˛()}.
E
1 1 Y (, t + 1) − Y (, t)
=
E
1 Y (, t)f
=
⎛ =
= ≥
⎝e
e
˛() − ˇ
˜ ˜ ˇ ˜ (˜ ) ˛( )−
−1 E
g
=E
E
0≤v≤t+1
≥
= =
0≤v≤t
sup Y (v, t)Y (t, t + 1)
E E
sup e{˛()(t−v)−S(v,t)}+{˛()−S(t,t+1)} − sup Y (v, t)
E
0≤v≤t
=
f (Y (1), . . ., Y (t))
0≤v≤t
0≤v≤t
e˛()−S(t,t+1) − 1
− sup Y (v, t)
f (Y ∗ (1), . . ., Y ∗ (t))
f (Y ∗ (1), . . ., Y ∗ (t))
0≤v≤t
E [Y ∗ (t)f (Y ∗ (1), . . ., Y ∗ (t))]
e˛()−ˇ() − 1 E [Y ∗ (t)f (Y ∗ (1), . . ., Y ∗ (t))] = 0 at = ∗
The inequality at the third step is due to the reduction of the upper bound of the supremum from t + 1 to t. The last equality is from the definition of * . This proves that Y* (t) is a demisubmartingale. To prove that Z(, t) is demisubmartingale with respect to at = * for > 0, 0 ≤ ≤ t, we need to show that for every coordinatewise non-decreasing, non-negative function f whenever
∗
sup Y (v, u) > e 0≤v≤u≤t
∗
∗
=P
0≤v≤u − ∗
sup (Y (u)) > e 0≤u≤t
− ∗
=E
sup
1 Y (v, t)
e−
∗
0≤v≤t
(11)
∗
(by Doob s maximal inequality)
sup Y (v, t) e
sup 0≤v≤u≤t
∗
=P
1
Y (t)
E
(12)
e−
∗
¨ (by applying Holder s inequality)
The first inequality is due to Doob’s inequality for demisubmartingales (cf. Corollary 2.4 in [7]). Application of Hölder’s inequality to Y1/ (cf. Theorem 3.6. especially equation (3.10) in [18]) provides the second inequality. From Eq. (12), the probability of interest is also bounded by the following limit.
P
∗
>e
0≤v≤t
sup Y (v, u)
−1
>
sup e{˛()(t+1−v)−S(v,t+1)} − sup Y (v, t)
≤ E [Y ∗ (t)] e
f (Y ∗ (1), . . ., Y ∗ (t))
0≤v≤t
(10)
sup
0≤v≤t+1
1 E [St ] x
˛( ∗ )(u − v) − S(v, u)
0≤u≤t
≤
=P
E[(Y ∗ (t + 1) − Y ∗ (t))f (Y ∗ (1), . . ., Y ∗ (t))]
=
sup
for ˜ =
≤
sup Sk > x
0≤v≤u≤t
sup Y (v, t + 1) − sup Y (v, t)
⎞
1 1 − 1⎠ E g Y (, + 1), . . ., Y (, t)
e˛()−ˇ() − 1 E [Z(, t)f (Z(t − 1, t), . . ., Z(, t))] = 0 at = ∗
P
0 at ˜ = ∗
E
For S(t), ˛( * ) and any > 0, we then get
Second and third equalities are due to the fact that the process Y(t) has stationary and independent increments and E[e−S(t,t+1) ] = e−ˇ() (cf., Eq. (6)), respectively. The final inequality ˜ = ˛(), ˜ ˜ > 0} and from the is due to the definition of ∗ = { : ˇ() ˜ ≥ ˛() ˜ for > 1. property of effective bandwidth cf., [16], i.e., ˛( ) This proves that Y1/ (, t) is a demisubmartingale.
=
E eA(−1,) E e−S(−1,) − 1 E [Z(, t)f (Z(t − 1, t), . . ., Z(, t))]
1 1 Y (, + 1), . . ., Y (, t)
1 1 Y (, + 1), . . ., Y (, t)
In the sequel we will make use ofDoob’s maximal inequality for demisubmartigales Sk , k = 1, . . ., t [7]
Second and third equalities are due to the fact that the process Z(t) has stationary and independent increments and E[e−S(−1,) ] = e−ˇ() , E[eA(−1,) ] = e˛() (cf., Eqs. (6), (5)), respectively. The final equality is due to the definition of * . This proves that Z(, t) is a demisubmartingale with respect to .
1 1 Y (, + 1), . . ., Y (, t)
=
eA(−1,)−S(−1,) − 1 Z(, t)f (Z(t − 1, t), . . ., Z(, t))
1≤k≤t
1 ˛() − S(t, t + 1)) − 1 E Y (, t)f E e
eA(−1,t−d)−S(−1,t) − eA(,t−d)−S(,t) f (Z(t − 1, t), . . ., Z(, t))
=
P
1 1 Y (, + 1), . . ., Y (, t)
f
1 Y (t, t + 1) − 1
E
= E
0≤v≤t
Proof. To prove that X(, t), Y(, t), Y1/ (, t) and Y* (t) are demisubmartingales with respect to t ∈ N = {1, 2, . . .} at = * for > 0, 0 ≤ ≤ t and any > 1, we need to show that for every coordinatewise non-decreasing, non-negative function f whenever the expectation is defined E[(X(, t + 1) − X(, t))f(X(, + 1), X(, + 2), . . ., X(, t))] ≥ 0, corresponding statements for Y(, t), Y1/ (, t) and Y* (t) will hold for t ∈ N = {1, 2, . . .}, 0 ≤ ≤ t and any > 1. As the proof for X(, t) and Y(, t) follow the same lines as Y1/ (, t) for = 1, we will provide the proofs only for Y1/ (, t) and Y* (t).
695
˛( ∗ )(u − v) − S(v, u)
≤ eE [Y (t)] e− = ee−
∗
∗
, as → ∞,
>
−1
, as E [Y (t)] = 1 at = ∗
→e (13)
The proof of Theorem 3.1 also relies on Lemma 4.1 from [14], which states that for any two non-negative independent random variables F and G with P(F > ) ≤ f() and P(G > ) ≤ g() where f() and g() are non-negative, decreasing functions for any ≥ 0, then
P F +G>
f˜ ( − u)dg˜ (u)
≤1−
(14)
0 −
where f˜ () = 1 − [f ()] , g˜ () = 1 − [g()]− and [a]− = min(1, a) for any a ≥ 0. Proof of Theorem 3.1 : We now provide the proof for the probabilistic end-to-end delay bound. For single hop (H = 1), the proof
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K. Angrishi, U. Killat / Int. J. Electron. Commun. (AEÜ) 69 (2015) 693–698
is straightforward and can be shown for a fixed sample path, t ≥ 0 and > 0 as follows:
P
=P
W (t) > d
= P ≤
sup
e
A(t − d) − D(t) > 0
∗ A(u,t−d)− ∗ S (u,t) 1
0≤u≤t
E e
∗ {A(t−d)−S (t)} 1
= e
≤P
time. However, the results can also be used in continuous time: Lemma 3.2 can be proven with any discretization step 0 and therefore is not limited to discretization step 0 = 1. Furthermore, as indicated in [19] Chapter 5.5, Doob’s maximal inequality and the use of Hölder’s inequality in the derivation of Eq. (13) also applicable for continuous time demisubmartingales and hence our result of Eq. (16) also applies to time-continuous problems. To analyse the accuracy of the new probabilistic end-to-end delay bound from Theorem 3.1, we compare it with the existing probabilistic bounds from network calculus and results from queueing theory for a network of M/M/1 queues. The customers arrive at rate and the server works at rate . We denote the utilization factor by = /, and assume for stability that < 1. The effective bandwidth and effective capacity of the arrival and service processes (Poisson process) are (e − 1)/ and (1 − e− )/, respectively, with = − log . We consider a special case of the network from Fig. 1 with H M/M/1
A(t − d) − A ⊗ S1 (t) > 0
(15)
>1
∗ ˛( ∗ )(t−d)− ∗ ˇ( ∗ )t
= e−
∗ ˛( ∗ )d
The first inequality is from the definition of stochastic network service process from Eq. (1). The final inequality is due to Doob’s inequality for demisubmartingale (Z(, t) with respect to ) from Eq. (10). The last two steps are due to our assumption about arrival A(t) and service S(t) processes having independent increments and from the definition of * , respectively. One can also arrive at the same result for H = 1 using a node operating point approach as described in [3]. In a multi-hop network (H > 1), for a fixed sample path, t ≥ 0 and > 0, we have, P
W (t) > d
= P
=P
A(t − d) − D(t) > 0
A(t − d) − A ⊗ Snet (t) > 0
A(t − d) − A ⊗ S1 ⊗ · · · ⊗ SH (t) > 0
= P
{A(k1 , t − d) −
sup
H−1
0≤k1 ≤k2 ≤···≤kH ≤t
≤
≤P
P
˛(h∗ )(kh+1
− kh ) −
{A(k1 , t − d) − ˛( )(t − d − k1 ) +
0≤k1 ≤k2 ≤···≤kH ≤t
≤
P
sup
≤
e
˛(H∗ )(t
− kH ) − SH (kH , t) +
A(k1 , t − d) − ˛( ∗ )(t − d − k1 )
+ sup 0≤kH ≤t
H h ( ∗ ˛( ∗ )d − (H − 1))
h!
The first inequality at the second step is from the definition of stochastic network service process from Eq. (1). The inequality at the fifth step is obtained by choosing ∗ = min h∗ ; note that h∗ 1≤h≤H
is the value of h at node h for which the effective bandwidth ˛ and the effective capacity ˇh satisfy the condition ˛(h∗ ) = ˇh (h∗ ). The justification for this equation lies in the (approximate) invariance of the effective bandwidth ˛() [4]. The inequality at the sixth step is from a property of supremum operation2 [14]. We get the final inequality from Lemma 3.2, (11), (13), (14) and by observing ∗ ∗ − ∗ ˛( ∗ ) that e h h ≤ e− ˛( ) holds for 1 ≤ h ≤ H. The proof is obtained by a complete induction over H. The final expression is valid only for ˛( * )d ≥ (H − 1)/ * . This constraint is a consequence of the [a]− operation in Eq. (14). The proof for the probabilistic bound on endto-end backlog is its immediate variation. It can be observed that setting H = 1 in Eq. (16), one gets the ∗ ∗ probability bound to be (1 + ∗ ˛( ∗ )d)e− ˛( )d . Though the bound is valid, it can be easily verified that it is worse than the bound ∗ ∗ e− ˛( )d from Eq. (15). This discrepancy between the two bounds is due to the fact that for the bound from Eq. (15) the stochastic process sup {A(u, t − d) − S1 (u, t)} is directly used to determine 0≤u≤t−d
the delay bound, in contrast to the bound from Eq. (16) where the arrival process with effective bandwidth and service process with effective capacity are used individually to establish the result. 4. Experiments using the new delay bound Complying with most of the relevant literature in this field we have based our derivation using demisubmartingales on discrete
sup {X(s) + Y (s)} ≤ sup {X(s)} + sup {Y (s)}. 0≤s≤t
0≤s≤t
H−1
˛(h∗ )(kh+1
˛(h∗ )(kh+1
− kh ) − Sh (kh , kh+1 ) } > 0
h=1
0≤s≤t
H−1
˛(H∗ )(t − kH ) − SH (kH , t)
h=0
2
− kH ) − SH (kH , t) +
H−1
(16)
∗
− kh ) − Sh (kh , kh+1 ) } > ˛( )d
h=1
0≤k1 ≤t
−( ∗ ˛( ∗ )d−(H−1))
− kH ) +
˛(H∗ )(t
h=1 ∗
sup
˛(H∗ )(t
+
h=1
sup
0≤kh ≤kh+1 ≤t
˛(h∗ )(kh+1 − kh ) − Sh (kh , kh+1 )
> ˛( ∗ )d
queues connected in series to analyse the accuracy of our endto-end network calculus delay bound. A Poisson flow with rate traverses through the entire network. Let each queue in the network be served by a service process S with effective capacity ˇ() and let the service processes at all nodes of the network be independent of each other. It is known from queueing theory that the exact distribution of steady state end-to-end delay of the through flow W(t) in a M/M/1 queueing network is given by
P W (t) > d
=
H−1 h ((1 − )d)
h!
e−(1− )d
(17)
h=0
The equation is obtained from an H-fold convolution of the (exponential) probability density function of delay for a single M/M/1 node, followed by an integration in the limits from d to infinity. The best available end-to-end delay bound W(t) from network calculus [8] in discrete time domain with discretization step 0 = 1 for the through flow in a network of tandem queues with both the arrival and the service processes having stationary and independent increments is given by
P W (t) > d
= inf
0≤≤ ∗
1 1 − e−{ˇ()−˛()}
H
e−˛()d
(18)
Eq. (17) has been derived under steady state conditions. Eqs. (16) and (18) hold for all values of t. As this includes very large values of t the bounds also address steady state situations. In Fig. 2, we illustrate the violation probability of the delay bounds (d = 4.5 ms and = 25 kbps) from Eqs. (8) and (9) as curve (b) along with the exact results from Eq. (17) as curve (a) and existing delay bounds from stochastic network calculus using moment generating functions [11,8] from Eq. (18) as curve (c) for a fixed utilization factor = 0.7 at each of the H queues. It can be observed that the new results provide better bounds than the existing bounds and also follow the shape of the exact results from queueing theory.
K. Angrishi, U. Killat / Int. J. Electron. Commun. (AEÜ) 69 (2015) 693–698
The above condition can be expressed in terms of effective bandwidth function for any t ≥ 0 and > 0, as shown below:
0
10
Violation Probability
1 (t − s)
∞
P(eA(s,t) > z)dz ≤ 0
−5
1 (t − s)
∞
P(eG(s,t) > z)dz 0
(20)
˛A (, t − s) ≤ ˛G (, t − s)
10
Similarly, we need a random process B with stationary and independent increments to stochastically lower bound the actual random service process S, i.e., B(s, t) ≤ st S(s, t), to enable us to employ the new probabilistic end-to-end delay bound from Theorem 3.1 for the connection admission control. The condition from Eq. (19) can be expressed in terms of effective capacity function for all t ≥ s ≥ 0 and > 0, as shown below:
−10
10
a) Exact result b) Bound using Theorem 3.1 c) Existing bound from network calculus −15
10
697
1
2
3
4
5
6
7 8 9 10 11 12 13 14 15 16 Number of Hops (H)
1 − (t − s)
∞
P(e
B(s,t)
0
P(A(s, t) > z) ≤ P(G(s, t) > z)
(19)
∞
P(eS(s,t) > z)dz 0
(21) Let the through and the cross arrival flows be modelled as an aggregate of Markov modulated On-Off arrival processes. This type of process has been used to model VoIP [1]. Markov modulated onoff process can be in “On” state or “Off” state for a random time interval which is negative exponentially distributed with average E[Ton ] and E[Toff ], respectively. In “On” state, arrival traffic transmits data at a constant rate R and no data is transmitted in “Off” state. This process is stochastically upper bounded by a process with stationary and independent increments which results from the original process by taking the time independent version of effective bandwidth ˛() = lim ˛(, t). Thus this derived process t→∞
can take the role of G in Eq. (20). ˛() for any > 0 is given by [9] ˛() =
1 (R − r10 − r01 + 2
2
(R − r10 + r01 ) + 4r10 r01 )
where r10 = 1/E[Ton ] and r01 = 1/E[Toff ]. Similarly, the leftover service process S at each network node h determined by general blind multiplexing model [11] is given by S(t) = Ct − AM (t) and there exists a random process B with stationary and independent increments, whose effective capacity is ˇh () = C − M˛() which stochastically lower bounds the leftover service process S at each network node. Now that we have two random processes with stationary and independent increments to describe the arrival and the service processes, respectively, we can use Eq. (8) in Theorem 3.1 and Eq. (18)
320 Bound using Theorem 3.1 Existing bound from network calculus Simulation results
300
Number through flows (N)
To illustrate the practical application of the new probabilistic end-to-end delay bound from Theorem 3.1, we employ it to perform connection admission control in a data network as shown in Fig. 3. An aggregate of N independent through arrival flows AN traverses H nodes connected in tandem and at each network node the through arrival flows share the resources with an aggregate of M independent cross arrival flows AM . Each network node is assumed to have a FIFO queue of infinite size and is served at a constant service rate C. The network is assumed to be stable, i.e., the capacity C is greater than the average arrival rate at each node. The simplified network in Fig. 3 used for the connection admission control can be viewed as segment of a possibly larger feed-forward network. There are some algorithms to turn non feed-forward networks to Feed-Forward Networks, namely, spanning tree and turn prohibition algorithm [21,10]. However, two critical assumptions are made regarding the flow path, namely, (i) arrival flows follow the same path for the entire duration of their transmission, (ii) effective bandwidth description of the cross flows at each network node along the path of through flow is known. The goal is to perform connection admission control of the through arrival flows AN in the tandem data network shown in Fig. 3 for a given probabilitic end-to-end delay bound with a fixed number M of cross arrival flows AM at each network node for different number of hops H in the tandem network. The main issue in the application of the new probabilistic end-to-end delay bound from Theorem 3.1 is its restriction to the networks with arrival and service processes having stationary and independent increments. However, if there exists a random process G with stationary and independent increments which stochastically upper bounds the actual random arrival process A, we can employ the random process G with stationary and independent increments for the analysis instead of the random arrival process A. A random arrival process A is said to be stochastically upper bounded by another random process G, and we write this as A(s, t) ≤ st G(s, t), if for all real z and all t ≥ s ≥ 0 the following condition is satisfied.
ˇB (, t − s) ≤ ˇS (, t − s)
Fig. 2. Violation probability of the end-to-end delay bound (d = 4.5 ms and = 25 kbps) in an M/M/1 network as a function of the number of nodes; utilization factor at each node is = 0.7.
Fig. 3. Network of H concatenated nodes with cross and through arrival flows.
1 > z)dz ≤ − (t − s)
280
260
240
220
200
180
2
4
6
8
10
12
14
16
Number of hops (H) Fig. 4. Maximum number of through flows allowable in the network for end-to-end delay bound of 300 ms with violation probability of 10−6 as a function of the number of nodes.
698
K. Angrishi, U. Killat / Int. J. Electron. Commun. (AEÜ) 69 (2015) 693–698
to perform connection admission control in a network as shown in Fig. 3. In the example, we determine the maximum number of through flows N allowable in the network without violating the probabilistic end-to-end delay bound of 300 ms with the violation probability of 10−6 . The capacity of the server C at each hop is set to 10 Mbps and the number of cross flows M at each hop set to a constant of 100 flows. To model a voice source using Markov modulated On-Off flows we set the parameters for each flow E[Toff ] = 650 ms, E[Ton ] = 352 ms and peak rate R = 64 kbps. Each Markov modulated On-Off flow produces data at an average rate m = 22.48 kbps. Fig. 4 shows the maximum number of through flows N allowable in the network without violating the probabilistic end-to-end delay bound of 300 ms with the violation probability of 10−6 as a function of increasing number of hops H. It can be seen that the new probabilistic delay bounds from Eqs. (8) and (9) provide better results than the existing network calculus result on probabilistic delay bound from Eq. (18). The new delay bounds benefit from the property of a sum of independent random variables defined in Eq. (14) and the demimartingales bounds, whereas the existing results from Eq. (18) provide loose delay bounds due to the use of union bounds. The simulation results give an upper bound for N determined from the first observation of a violation of the given delay bound of 300 ms 5. Conclusions In this paper we used demimartingale inequalities to compute end-to-end probabilistic delay and backlog bounds within the framework of network calculus. The tightness of the computed end-to-end probabilistic performance bounds is explored by comparing new bounds with the exact results from queueing theory for a network of M/M/1 queues. References [1] International Telecommunication Union. Recommendation ITU-T P.59, Artificial Conversational Speech; 1993. [2] Angrishi K. An end-to-end stochastic network calculus with effective bandwidth and effective capacity. Comput Netw 2013;57(1):78–84.
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