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... random variables X and Y , we say X is stochastically equal to (smaller than; larger than) Y , written X =st (≤st; ≥st)Y , if P {X > x} = (≤; ≥)P {Y > x} or P {X ≤ x} = (≥; ≤)P {Y ≤ x} for all x [35]=-=[30]-=-. We denote by F the set of non-negative wide-sense increasing functions, or F = {f(·) : ∀0 ≤ x ≤ y,f(x) ≥ 0,f(x) ≤ f(y)}, and by ¯F the set of non-negative wide-sense decreasing functions, or, ¯F = {...

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...: deterministic traffic models and stochastic traffic models. The (σ, ρ) or token bucket constrained traffic model is perhaps the most widely studied deterministic one due to the seminar work of Cruz =-=[1]-=-. The (σ, ρ) model has been generalized to the concept of arrival envelope or arrival curve for deterministic service guarantee analysis [2][3]. Because deterministic service guarantee can result in (...

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... arrival curve (also called envelope in other literatures, e.g. [7]) as follows [3]: Definition 2. A flow is said to have an arrival curve α(∈ F) iff for all 0 ≤ s ≤ t, there holds A(s, t) ≤ α(s, t). =-=(3)-=- By letting α(s, t) =σ + ρ(t − s) in (3), the arrival curve model is reduced to the (σ, ρ) model. Based on the concept of arrival curve and the concept of service curve, many results have been derived...

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...s stochastic ordering between two random variables. Specifically, random variable X is said to be stochastically smaller than random variable Y , denoted by X ≤st Y , if for any x, P {X >x}≤P {Y > x} =-=[13]-=-. Also in Theorem 2, by letting P {σ >x}≤f(x) for the random variable σ, the first part, i.e. A(s, t) − α(s, t) ≤st σ, implies the definition of the t.a.c stochastic arrival curve, and the second part...

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...s ≤ u ≤ t. The convolution of two functions f and g, denoted by f ⊗ g, is defined as f ⊗ g(x)= min[f(y)+g(x − y)]. (1) 0≤y≤x 3If both f and g belong to F, (1) is the same as the (min, +) convolution =-=[3]-=- [15] and many properties of it have been proved [15]. These properties include: closure property, i.e. ∀f, g ∈F, f ⊗ g ∈F; commutativity, i.e. ∀f, g ∈F, f ⊗ g = g ⊗ f; associativity, i.e. ∀f, g, h ∈F...

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...e analysis has been identified as a grand challenge for future networking research [36]. Towards it, stochastic network calculus, the probabilistic version of the (deterministic) network calculus [11]=-=[12]-=-[13][6][21] [1] [22][9], has been recognized by researchers as a crucial step (e.g. [36][25]). 1In recent years, many attempts have been made for the development of a stochastic network calculus, whi...

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...u,s) − r(s − u)] [A(s − u) − r(s − u)]. (50) While (50) seems to be obvious since A(u,s) is stationary, it actually follows from a non-trivial result on stochastic ordering, which is Theorem 2.2.3 of =-=[35]-=-. The right hand side of (50) can be simplified as: with which and (50), (49) is proved. sup sup 0≤s≤t 0≤u≤s = sup ≤ 0≤s≤t 0≤v≤s sup 0≤s≤t v≥0 [A(s − u) − r(s − u)] sup [A(v) − r · v] sup[A(v) − r · v...

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...al curve α ∈F with bounding function f ∈ ≼ F, denoted by A ∼vb 〈f, α〉, iff for all t ≥ 0 and all x ≥ 0, there holds P { max {A(s, t) − α(t − s)} >x}≤f(x). (4) 0≤s≤t The following result introduced in =-=[20]-=- [5] [12] can be used to find the v.b.c stochastic arrival curve of a flow: Lemma 2. Suppose a(t) ≡ A(t) − A(t − 1) is stationary and ergodic. Then, if E{a(1)} <r, there holds, for all t ≥ 0 and x ≥ 0...

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...y studied deterministic one due to the seminar work of Cruz [1]. The (σ, ρ) model has been generalized to the concept of arrival envelope or arrival curve for deterministic service guarantee analysis =-=[2]-=-[3]. Because deterministic service guarantee can result in (possibly very) low network resource utilization and because many applications and networks do not require or provide deterministic ⋆⋆ “Centr...

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...alysis has been identified as a grand challenge for future networking research [36]. Towards it, stochastic network calculus, the probabilistic version of the (deterministic) network calculus [11][12]=-=[13]-=-[6][21] [1] [22][9], has been recognized by researchers as a crucial step (e.g. [36][25]). 1In recent years, many attempts have been made for the development of a stochastic network calculus, which i...

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...d I(0) = 0 by convention. It is clear that S, I are in F and additive. We now define strict stochastic server as follows: Definition 6. A server S is said to be a strict stochastic server providing strict stochastic service curve β(·) ∈ F with impairment process I to a flow iff during any backlogged period (s, t], the output A∗(s, t) of the flow from the server satisfies A∗(s, t) ≥ β(t − s) − I(s, t). In the rest, we assume β is additive and has the form of β(t) = rt. In addition, we assume the impairment process I(t) is (σ(θ), ρ(θ))-upper constrained, a model that was initially used in [4] to characterize stochastic behavior of traffic, whose definition is as follows: Definition 7. A stochastic sequence I, I ≡ {I(t), t = 0, 1, 2, . . .} with I(0) = 0, is said to be (σ(θ), ρ(θ))-upper constrained (for some θ > 0), iff for all 0 ≤ s ≤ t 1 θ logEeθ(I(t)−I(s)) ≤ ρ(θ)(t − s) + σ(θ). (20) The following result shows that if the impairment process I(t) is (σ(θ), ρ(θ))- upper constrained, a strict stochastic server has a stochastic service curve. Due to space limitation, the proof is omitted and can be found from [11]. Theorem 5. Consider a strict stochastic server providing strict stoc...

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...he required sequences of random variables. Another type is built upon moments or moment generating functions. This type was initially used for traffic [6][19] and has also been extended to service [8]=-=[37]-=-. Independence assumption is generally required between arrival and service processes. Extensive study has been conducted for deriving the characteristics of a process under this type of model from so...

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... [1] [22][9], has been recognized by researchers as a crucial step (e.g. [36][25]). 1In recent years, many attempts have been made for the development of a stochastic network calculus, which include =-=[20]-=-[38][6][7][23][14][29][33][39][5][24][26][3][18][17][10]. However, the following basic properties required by a network calculus have made the stochastic network calculus challenging: (P.1) (Superposi...

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...any stochastic traffic models have been proposed in the literature for stochastic service guarantee analysis, most of them can be considered as special cases of Definition 3 as demonstrated below. In =-=[10]-=-, a stochastic traffic model, called exponentially bounded burstiness (EBB), is proposed, which is later extended in [4] to the stochastically bounded burstiness (SBB) model. Both EBB and SBB are spec...

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...arrival curve A(t) ∼vb 〈f Gaussian ,rt〉 with f Gaussian � (x +(r − ˆr)s)2 (x) =exp − 2ˆv ∗ � , (9) reaches its minimum at s ∗ . where ˆv ∗ ≡ ˆv(s ∗ ) and s∗ is chosen such that (x+(r−ˆr)s)2 2ˆv(s) In =-=[17]-=- [2], a stochastic traffic model (σ(θ),ρ(θ)), called θ-MER (minimum envelope rate with respect to θ), is proposed for studying stationary and ergodic traffic. It is shown in [17] and [2] that many typ...

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...sed on this property is intuitively simple, recent study has revealed that the applicability of using these resulting stochastic traffic models to stochastic service guarantee analysis is limited. In =-=[4]-=- [5], some restriction is enforced on the proposed stochastic traffic model itself to ensure that the resulting one can be used for deriving stochastic service guarantees. In [6], the problem of using...

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...een identified as a grand challenge for future networking research [36]. Towards it, stochastic network calculus, the probabilistic version of the (deterministic) network calculus [11][12][13][6][21] =-=[1]-=- [22][9], has been recognized by researchers as a crucial step (e.g. [36][25]). 1In recent years, many attempts have been made for the development of a stochastic network calculus, which include [20]...

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...alysis is limited. In [4] [5], some restriction is enforced on the proposed stochastic traffic model itself to ensure that the resulting one can be used for deriving stochastic service guarantees. In =-=[6]-=-, the problem of using its proposed stochastic traffic model to analyze stochastic service guarantees is highlighted. Although an approach is proposed in [6] to fix the problem, the consequence is tha...

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...with Poisson traffic. Suppose each packet of a flow has the same normalized size and packet arrivals of the flow follow a Poisson process with mean arrival rate λ. Then, based on M/D/1 analysis (e.g. =-=[14]-=-), the flow has a v.b.c stochastic arrival curve A(t) ∼vb 〈f P oisson ,rt〉 for any r>λ, where f P oisson (x) =1− (1 − λ r ) ⎡ x� λ(m−x) ( ) ⎣ r m=0 m ⎤ λ(m−x) − e r ⎦ . (8) m! We next consider Gaussia...

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... recognized by researchers as a crucial step (e.g. [36][25]). 1In recent years, many attempts have been made for the development of a stochastic network calculus, which include [20][38][6][7][23][14]=-=[29]-=-[33][39][5][24][26][3][18][17][10]. However, the following basic properties required by a network calculus have made the stochastic network calculus challenging: (P.1) (Superposition Property) The sup...

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...ucial step (e.g. [36][25]). 1In recent years, many attempts have been made for the development of a stochastic network calculus, which include [20][38][6][7][23][14][29][33][39][5][24][26][3][18][17]=-=[10]-=-. However, the following basic properties required by a network calculus have made the stochastic network calculus challenging: (P.1) (Superposition Property) The superposition of flows can be represe...

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...has been identified as a grand challenge for future networking research [36]. Towards it, stochastic network calculus, the probabilistic version of the (deterministic) network calculus [11][12][13][6]=-=[21]-=- [1] [22][9], has been recognized by researchers as a crucial step (e.g. [36][25]). 1In recent years, many attempts have been made for the development of a stochastic network calculus, which include ...

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...an arrival process. Let ˆrt and ˆv(t) respectively be the mean and variance of the Gaussian arrival process A(t). Then, available ( simulation and analytical results in the literature [7] (x+(r−ˆr)s) =-=[17]-=- [1][22] suggest that for all r>ˆr, exp −inf s≥0 2 ) , is an upper bound on P { ˆQ(t; r) > 2ˆv(s) x}. Hence, for this type of traffic, it also has v.b.c stochastic arrival curve A(t) ∼vb 〈f Gaussian ,...

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...under the context of network calculus, most traffic models used in the literature [38][14][33][39][5][24][26][3][18][17][10] belong to t.a.c and v.b.c stochastic arrival curve, and most server models =-=[23]-=-[14][5][24][26][3][10] belong to weak stochastic service curve. Table 1 shows that without additional constraints, these works can only support part of the five required properties for the stochastic ...

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...by researchers as a crucial step (e.g. [36][25]). 1In recent years, many attempts have been made for the development of a stochastic network calculus, which include [20][38][6][7][23][14][29][33][39]=-=[5]-=-[24][26][3][18][17][10]. However, the following basic properties required by a network calculus have made the stochastic network calculus challenging: (P.1) (Superposition Property) The superposition ...

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...xt consider Gaussian arrival process. Let ˆrt and ˆv(t) respectively be the mean and variance of the Gaussian arrival process A(t). Then, available simulation and analytical results in the literature =-=[15]-=- [16] suggest that for all r > ˆr, exp � (x+(r−ˆr)s) −inf s≥0 2 � , is an upper 2ˆv(s) bound on P {W (t; r) >x}. Hence, for this type of traffic, it has a v.b.c stochastic arrival curve A(t) ∼vb 〈f Ga...

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...been recognized by researchers as a crucial step (e.g. [36][25]). 1In recent years, many attempts have been made for the development of a stochastic network calculus, which include [20][38][6][7][23]=-=[14]-=-[29][33][39][5][24][26][3][18][17][10]. However, the following basic properties required by a network calculus have made the stochastic network calculus challenging: (P.1) (Superposition Property) The...

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...e network case, the network delay and backlog can be a function of f (n) (x) where n is the number of nodes along the path of a flow. When n increases, f (n) (x) can become too large to be meaningful =-=[11]-=-. Since FIFO is the simplest scheduler for implementing a network node, we hence believe it is difficult to apply the stochastic traffic model defined in Definition 3 to perform stochastic service gua...

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... of this, additional conditions on the process may be needed to ensure that M(t) converges in distribution. Such conditions have been discussed in the literature for the study of maximum queue length =-=[2]-=-[32][31]. In the rest, we consider a condition that has been widely used (e.g [6][8]). The condition is that a(t) is a sequence of independent identically distributed (i.i.d.) random variables. Here, ...

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...uch as Markov Modulated Processes have been proved to be (σ(θ),ρ(θ))-upper constrained. Note that these processes have also been used in the literature for characterizing a wireless channel (e.g. [9] =-=[10]-=-). We hence believe our results are useful for stochastic service guarantee analysis in such networks. 5 Conclusion In this paper, we introduced a new server model, called stochastic service curve, fo...

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...ed on M/D/1 analysis (e.g. [14]), the flow has a v.b.c stochastic arrival curve A(t) ∼vb 〈f P oisson ,rt〉 for any r>λ, where f P oisson (x) =1− (1 − λ r ) ⎡ x� λ(m−x) ( ) ⎣ r m=0 m ⎤ λ(m−x) − e r ⎦ . =-=(8)-=- m! We next consider Gaussian arrival process. Let ˆrt and ˆv(t) respectively be the mean and variance of the Gaussian arrival process A(t). Then, available simulation and analytical results in the li...

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...ion set G for which, any f ∈ G implies its n−fold integration f (n) also belongs to G. However, some traffic models may not have bounding functions in G. One example is the α−stable traffic model [19]=-=[20]-=-. The α−stable is a traffic model characterizing the self-similar behavior of traffic [19][20]. The model is defined by four parameters: (α,H,c1,c2). In [19], it is shown that the queue length � �−α r...

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...], has been recognized by researchers as a crucial step (e.g. [36][25]). 1In recent years, many attempts have been made for the development of a stochastic network calculus, which include [20][38][6]=-=[7]-=-[23][14][29][33][39][5][24][26][3][18][17][10]. However, the following basic properties required by a network calculus have made the stochastic network calculus challenging: (P.1) (Superposition Prope...

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...own traffic and service characterizations to the required sequences of random variables. Another type is built upon moments or moment generating functions. This type was initially used for traffic [6]=-=[19]-=- and has also been extended to service [8][37]. Independence assumption is generally required between arrival and service processes. Extensive study has been conducted for deriving the characteristics...

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...nsider Gaussian arrival process. Let ˆrt and ˆv(t) respectively be the mean and variance of the Gaussian arrival process A(t). Then, available simulation and analytical results in the literature [15] =-=[16]-=- suggest that for all r > ˆr, exp � (x+(r−ˆr)s) −inf s≥0 2 � , is an upper 2ˆv(s) bound on P {W (t; r) >x}. Hence, for this type of traffic, it has a v.b.c stochastic arrival curve A(t) ∼vb 〈f Gaussia...

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...rival process. Let ˆrt and ˆv(t) respectively be the mean and variance of the Gaussian arrival process A(t). Then, available ( simulation and analytical results in the literature [7] (x+(r−ˆr)s) [17] =-=[1]-=-[22] suggest that for all r>ˆr, exp −inf s≥0 2 ) , is an upper bound on P { ˆQ(t; r) > 2ˆv(s) x}. Hence, for this type of traffic, it also has v.b.c stochastic arrival curve A(t) ∼vb 〈f Gaussian ,rt〉 ...

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...tified as a grand challenge for future networking research [36]. Towards it, stochastic network calculus, the probabilistic version of the (deterministic) network calculus [11][12][13][6][21] [1] [22]=-=[9]-=-, has been recognized by researchers as a crucial step (e.g. [36][25]). 1In recent years, many attempts have been made for the development of a stochastic network calculus, which include [20][38][6][...

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...zation of deterministic network calculus has been considered as an important step and several attempts have been made [23] [16] [21] [11] [4][17][19][2] 1 . 1 Recently, there are several attempts [7] =-=[1]-=- studying time-varying systems using the (min, +) algebra. The difference between these attempts and the work in this paper is that although time-varying, the studied servers in the former have determ...

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...rver model satisfying (P.1) - (P.4) has been available in the literature. The most widely used one, which we shall call weak stochastic service curve, was introduced by Cruz [11]. Although authors in =-=[19]-=- have adopted weak stochastic service curve as the server model and derived interesting results for stochastic service guarantee analysis, the weak stochastic service curve model, while having propert...

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...on this property is intuitively simple, recent study has revealed that the applicability of using these resulting stochastic traffic models to stochastic service guarantee analysis is limited. In [4] =-=[5]-=-, some restriction is enforced on the proposed stochastic traffic model itself to ensure that the resulting one can be used for deriving stochastic service guarantees. In [6], the problem of using its...

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...d to a flow are investigated. These analyses make valuable contribution to network calculus, because, although deterministic service guarantee analysis has been conducted for the time-varying setting =-=[7]-=-, no result has been available for stochastic service guarantee analysis under the time-varying setting. Finally, we present a stochastic ordering monotonicity property of the v.b.c stochastic arrival...

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...unction set G for which, any f ∈ G implies its n−fold integration f (n) also belongs to G. However, some traffic models may not have bounding functions in G. One example is the α−stable traffic model =-=[19]-=-[20]. The α−stable is a traffic model characterizing the self-similar behavior of traffic [19][20]. The model is defined by four parameters: (α,H,c1,c2). In [19], it is shown that the queue length � �...

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...twork satisfies, for all t ≥ 0 and all x ≥ 0, P {D(t) >x}≤ ⊗N n=1 hn (x). 4.2 A Numerical Example This subsection presents a numerical example to illustrate the use of the results developed above. In =-=[14]-=-, we proposed a dynamic priority scheduling (DPS) scheme to aggregate flows for per-flow measurement-based admission control (MBAC). In this DPS MBAC, each admitted flow has a dedicated queue; an earl...

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...h α−stable traffic can also be modeled using a v.b.c stochastic arrival curve with A(t) ∼vb 〈f Alpha ,rt〉 where f Alpha � � r − m −α (x) =Cα . (13) c1 The probabilistic burstiness curve (PBC) concept =-=[21]-=- uses a stationary random process to model real-time multimedia traffic. Particularly, it uses the steady-state queue distribution of a constant rate server fed with a flow to represent the traffic of...

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...s, additional conditions on the process may be needed to ensure that M(t) converges in distribution. Such conditions have been discussed in the literature for the study of maximum queue length [2][32]=-=[31]-=-. In the rest, we consider a condition that has been widely used (e.g [6][8]). The condition is that a(t) is a sequence of independent identically distributed (i.i.d.) random variables. Here, a(t) den...

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... x}. The proof of (8) can be found from the literature (e.g. [9][19][2]). With the monotonicity property of ⊗, if P {X >x}≤f(x) andP {Y >x}≤g(x), we get from (8) that P {X + Y > x}≤f ⊗ g(x). (9) 1 In =-=[1]-=-, service curve with loss is defined. It should be noticed that this definition is different from Definitions 4, 5 and 6 here. In a service curve with loss network element, packets are dropped if thei...

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...x) =f(x)+ 1 ɛ f (1) (x), (12) for any ɛ>0. A traffic model satisfying (12) is Weibull Bounded (WB) model that has commonly been used to characterize the long range dependent (LRD) behavior of traffic =-=[18]-=-. It is worth highlighting that traffic models reviewed above have bounding functions with exponential forms and/or belonging to the function set G for which, any f ∈ G implies its n−fold integration ...

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...hich include (i) Associativity: For any functions f, g, and h in F, (f ⊗ g) ⊗ h = f ⊗ (g ⊗ h); (ii) Commutativity: For any functions f and h in F, f ⊗ h = h ⊗ f. In addition, it has been proved (e.g. =-=[9]-=-) that for any n random variables Wi, i =1, 2,...,n,ifP {Wi >x}≤fi(x) for any x ≥ 0, then n� n� P { Wi >x}≤ fi(x), (2) i=1 i=1 where � n i=1 fi(x) ≡ f1 ⊗ f2 ⊗···⊗fn(x). For performance guarantee analy...

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...archers as a crucial step (e.g. [36][25]). 1In recent years, many attempts have been made for the development of a stochastic network calculus, which include [20][38][6][7][23][14][29][33][39][5][24]=-=[26]-=-[3][18][17][10]. However, the following basic properties required by a network calculus have made the stochastic network calculus challenging: (P.1) (Superposition Property) The superposition of flows...

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... this, additional conditions on the process may be needed to ensure that M(t) converges in distribution. Such conditions have been discussed in the literature for the study of maximum queue length [2]=-=[32]-=-[31]. In the rest, we consider a condition that has been widely used (e.g [6][8]). The condition is that a(t) is a sequence of independent identically distributed (i.i.d.) random variables. Here, a(t)...

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...search [25]. Towards it, stochastic network calculus, the probabilistic generalization of (min, +) (deterministic) network calculus [9][7][17], has been considered as a fundamental and important step =-=[20]-=-. Many challenges have made stochastic network calculus difficult. Some of them are closely related to server modeling, which include output characterization, concatenation property, stochastic backlo...