A Temporal Approach to Stochastic Network Calculus ¹¹1An early version of this paper was partially presented at MASCOTS 2009 [36].

We use $P(n)$, $a(n)$, $d(n)$ and $\delta_{n}$, to denote the $(n+1)th$ packet entering the system, its arrival time to the system, its departure time from the system and its service time provided by the system, respectively, where $n=0,1,2,...$.

• 1.

  From the temporal perspective, an arrival process counts the cumulative inter-arrival time between two arbitrary packets and is denoted by $\Gamma(m,n)=a(n)-a(m)$ for any $0\leq m\leq n$. Note $\Gamma(n,n)=0$.

• 2.

  A service process describes the cumulative service time received between two arbitrary packets and is denoted by $\Delta(m,n)=\sum_{k=m}^{n}\delta_{k}$ for any $0\leq m\leq n$. Note $\Delta(n,n)=\delta_{n}$.

In the time-domain, the system backlog and system delay are defined below, respectively.

Moreover, the time that packet $P(n)$ waits in queue is denoted by $W(n)$:

The following function sets are often used in this paper.

The set of non-negative wide-sense increasing functions is denoted by $\mathcal{F}$, where for each function $f(\cdot)$,

and for any function $f(\cdot)\in\mathcal{F}$, we set $f(x)=0$ for all $x<0$.

We denote by $\bar{\mathcal{F}}$ the set of non-negative wide-sense decreasing functions where for each function $f(\cdot)$,

and for any function $f(\cdot)\in\bar{\mathcal{F}}$, we set $f(x)=1$ for all $x<0$.

We denote by $\bar{\mathcal{G}}$ a subset of $\bar{\mathcal{F}}$, where for each function $f(\cdot)\in\bar{\mathcal{G}}$, its nth-fold integration, denoted by $f^{(n)}(x)\equiv(\int_{x}^{\infty}~dy)^{n}f(y)$, is bounded for any $x\geq 0$ and still belongs to $\bar{\mathcal{G}}$ for any $n\geq 0$, i.e.,

For ease of exposition, we adopt the following notations in this paper:

In addition, the ceiling and floor functions are used in this paper as well.

• 1.

  The ceiling function $\lceil x\rceil$ returns the smallest integer not less than $x$.

• 2.

  The floor function $\lfloor x\rfloor$ returns the larget integer not greater than $x$.

An essential idea of (stochastic) network calculus is to use alternate algebras, particularly min-plus algebra and max-plus algebra [5], to transform complex non-linear network systems into analytically tractable linear systems. To the best of our knowledge, the existing models and results of stochastic network calculus mainly based on min-plus algebra that has basis operations suitable for characterizing the amount of cumulative traffic and service. As a result, these models focus on describing network behavior from the spatial perspective. Max-plus algebra is suitable for arithmetic operations with cumulative inter-arrival times and service times. Consequently, network modeling from the temporal perspective more relies on max-plus algebra. In the following, we review the basics of both min-plus algebra and max-plus algebra.

In min-plus algebra, the ‘addition’ operation represents infimum or minimum when it exists, and the ‘multiplication’ operation is $+$. The min-plus convolution of functions $f,g\in\mathcal{F}$, denoted by $\otimes$, is defined as

where, when it applies, ‘infimum’ should be interpreted as ‘minimum’. The min-plus deconvolution of functions $f,g\in\mathcal{F}$, denoted by $\oslash$, is defined as

where, when it applies, ‘supremum’ should be interpreted as ‘maximum’.

In the max-plus algebra, the ‘addition’ operation represents supremum or maximum when it exists, and the ‘multiplication’ operation is $+$. The max-plus convolution of functions $f,g\in\mathcal{F}$, denoted by $\bar{\otimes}$, is defined as

where, when it applies, ‘supremum’ should be interpreted as ‘maximum’. The max-plus deconvolution of functions $f,g\in\mathcal{F}$, denoted by $\bar{\oslash}$, is defined as

where, when it applies, ‘supremum’ should be interpreted as ‘maximum’.

The max-plus convolution is associative and commutative [5].

• 1.

  Associativity: for any $g_{1},g_{2},g_{3}\in\mathcal{F}$, $(g_{1}\bar{\otimes}g_{2})\bar{\otimes}g_{3}=g_{1}\bar{\otimes}(g_{2}\bar{\otimes}g_{3})$.

• 2.

  Commutativity: for any $g_{1},g_{2}\in\mathcal{F}$, $g_{1}\bar{\otimes}g_{2}=g_{2}\bar{\otimes}g_{1}$.

The available literature on stochastic network calculus mainly focuses on modeling network behavior and analyzing network performance from the spatial perspective [6][12][15][16][20][22][23][28][29][31]. We call the corresponding models and results space-domain models and results in this paper.

In order to characterize the arrival process of a flow from the spatial perspective, let us consider the amount of traffic generated by this flow in a time interval $(s,t]$, denoted by $\mathcal{A}(s,t)$. In the context of stochastic network calculus, the arrival curve model is defined based on a stochastic upper bound on the cumulative amount of the arrival traffic. Here, we only review one relevant space-domain arrival curve model, virtual-backlog-centric ($v.b.c$) stochastic arrival curve (SAC) [22].

In stochastic network calculus, the service curve model is defined as a stochastic lower bound on the cumulative amount of service provided by the system. Two space-domain service curve models [22] are reviewed here.

Unlike the arrival curve, it is difficult to identify the service curve from (5) because it couples the arrival process, the service curve and the departure process. Thus we need a more explicit model to directly reveal the relation between the service process and its service curve such as the following model [20].

Definition 4 is applied to ‘any period’ which implies both worst-case scenario and other scenarios. If we could determine a function $\beta(t)$ which makes Eq.(6) hold under the worst-case scenario, then Eq.(6) automatically holds under other scenarios as well.

Based on the arrival curve and service curve models, five fundamental properties have been proved to facilitate tractable analysis. For example, they can be used to derive service guarantees including delay bound and backlog bound, characterize the behavior of traffic departing from a server, describe the service provided along a multi-node path, determine the arrival curve for the aggregate flow, and compute the service provided to each constituent flow.

• 1.

  P.1: Service Guarantees (single-node)

  Under the condition that the traffic arrival process has an arrival curve $\alpha(t)$ with bounding function $f(x)$ and the network node provides service with a service curve model $\beta(t)$ with bounding function $g(x)$, the stochastic delay bound and stochastic backlog bound can be derived. Particularly, the backlog bound is related to the maximal vertical distance between $\alpha(t)$ and $\beta(t)$; the delay bound is relevant to the maximal horizontal distance between $\alpha(t)$ and $\beta(t)$.

• 2.

  P.2: Output Characterization

  To analyze the end-to-end performance of a multi-hop path, one option is the node-by-node analysis approach. This approach requires being able to characterize the traffic behavior after the traffic has been served and leaves the previous node. The output process of a flow from a node can also be characterized by an arrival curve which is determined by both the arrival curve of the arrival process and the service curve of the service process.

• 3.

  P.3: Concatenation Property (multi-node)

  Network calculus possesses an unique property, concatenation property, which is also used to analyze the end-to-end performance but improves the results obtained from the node-by-node analysis. The essence of the concatenation property is to represent a series of nodes in tandem as a ‘black box’ which can be treated as a single node. The service curve of this equivalent system is determined by the service curve of all individual nodes along this path.

• 4.

  P.4: Superposition Property (aggregate flow)

  Flow aggregation is very common in packet-switched networks. If multiple flows are aggregated into a single flow under the FIFO order, the aggregate flow also has an arrival curve which is the summation of the arrival curve of all constituent flows.

• 5.

  P.5: Leftover Service Characterization (per-flow)

  The leftover service characterization makes per-flow performance analysis feasible under FIFO aggregate scheduling. The crucial concept is to represent all other constituent flows as an ‘aggregate cross flow’ which can be characterized using an arrival curve. Then the service provided to the constituent flow of interest can also be described by a service curve which is determined by the service curve provided to all arrival flows and the arrival curve of the ‘aggregate cross flow’.

The superposition property of the $v.b.c$ SAC [22] is reviewed here because it is relevant to the model transformation in the following content.

Consider an arrival process that specifies packets arriving to a network system at time $a(n)$, $n=0,1,2,...$. In order to stochastically guarantee a certain level of QoS to this arrival process, this arrival process should be constrained. By characterizing the constrained arrival traffic from the temporal perspective, we define an inter-arrival-time (i.a.t) stochastic arrival curve model.

Eq.(7) indicates that function $\lambda(n)$ is a probabilistic lower bound on the cumulative inter-arrival time. The violation probability that the cumulative inter-arrival time is smaller than $\lambda(n)$ is bounded above by function $h(x)$. If $h(x)=0$ for all $x\geq 0$, Eq.(7) represents a time-domain deterministic arrival curve [10] which is a special case of the $i.a.t$ SAC.

Queueing theory typically characterizes the arrival process using the probability distribution of the inter-arrival time between two consecutive customers:

Comparing $F(x)$ with Eq.(7), we notice that Eq.(7) gives a more general probability expression of the inter-arrival time between two arbitrary packets. From this viewpoint, $F(x)$ is a special case of Eq.(7).

Consider a flow of packets with fixed packet size. Suppose that packet inter-arrival times follow an exponential distribution with mean $1/\mu$. Then, the packet arrival time has an Erlang distribution with parameter $(n,\mu)$ [1], where $n$ denotes the number of arrival packets. For any two packets $P(m)$ and $P(m+n)$, their inter-arrival time satisfies, for $x\geq 0$,

where $y=\frac{n}{\mu}-x$. Thus, the flow has an $i.a.t$ SAC $\lambda(n)=\frac{n}{\mu}$.

The $i.a.t$ SAC is simple but has limited applications. For example, consider a virtual single server queue (SSQ) fed with the arrival traffic which has an $i.a.t$ SAC $\lambda(n)$ with bounding function $h(x)$. Suppose that the virtual SSQ provides a constant service time $\lambda(1)$ for each packet. From Eq.(3), the waiting delay of $P(n)$ experienced in the virtual SSQ is

where $a(m)$ is the beginning of the backlogged period within which packet $P(n)$ is transmitted. Eq.(8) is derived from the departure time given in Eq.(13). Eq.(9) is called the virtual-waiting-delay property. It is difficult to compute the virtual-waiting-delay from Eq.(7). When investigating the performance guarantees such as delay bound and backlog bound in Section 4.1, we face the similar difficulty.

In order to deal with the difficulty of computing the virtual-waiting-delay, we define another stochastic arrival curve model based on Eq.(9).

Through some manipulations, Eq.(10) can be expressed as the max-plus convolution:

Here, $a\bar{\otimes}\lambda(n)$ can be considered as the expected time that the packet would arrive to the head-of-line (HOL) if the flow has passed through a virtual SSQ with the (deterministic) service curve $\lambda(n)$. The packet is expected to arrive not earlier than the expected HOL time. Here $x$ represents the difference between the expected HOL time and the actual arrival time. The violation probability is bounded by the non-increasing function $h(x)$.

We use the $v.w.d$ SAC to characterize the arrival traffic in Example 1.

Consider a flow that consists of packets having the fixed packet size. Suppose that all packet inter-arrival times are exponentially distributed with mean $\frac{1}{\mu}$. Based on the steady-state probability mass function (PMF) of the queue-waiting time for an M/D/1 queue [33], we say that the flow has a $v.w.d$ SAC $\lambda(n)=\hbar\cdot n$ with bounding function $h^{exp}$ for $0<\hbar<\frac{1}{\mu}$. Let $\rho=\mu\cdot\hbar$. We obtain the bounding function of the probability that the waiting delay $W(n)$ exceeds $x(\geq 0)$

where, $\lfloor y\rfloor$ denotes the floor function.

The definition of $v.w.d$ SAC is more strict than that of $i.a.t$ SAC. As a result, it is not trivial to derive the $v.w.d$ SAC for an arrival process even if it can be characterized by an $i.a.t$ SAC. Thus, it is important to explore whether there exists some relationship between the $i.a.t$ SAC and the $v.w.d$ SAC.

Remark. In the second part, $h(x)\in\bar{\mathcal{G}}$ while not $h(x)\in\bar{\mathcal{F}}$. If the requirement on the bounding function is relaxed to $h(x)\in\bar{\mathcal{F}}$, the second part may not hold in general.

Theorem 2 reveals that if an arrival process can be modeled by a $v.w.d$ SAC $\lambda(n)$, then $\lambda(n)$ is also the $i.a.t$ SAC of this arrival process. On the other hand, if an arrival process can be modeled by an $i.a.t$ SAC with the associated bounding function in $\bar{\mathcal{G}}$, then this arrival process also has a $v.w.d$ SAC which may be associated with a more loose bounding function.

It is worth highlighting that the $v.w.d$ SAC looks similar to the $v.b.c$ SAC (see Definition 2) defined in the space-domain. Since these two models play an important role in performance analysis in their respective domains, we establish their relationship in the following theorem.

Note that in Theorem 3, the arrival curve $\alpha(t)$ denotes the cumulative number of arrival packets while not the cumulative amount (in bits) of arrival traffic.

The generalized stochastically bounded burstiness (gSBB) [38] is a special case of the space-domain $v.b.c$ SAC. A summarization of some well-known traffic belonging to gSBB is given [22], including both Gaussian self-similar processes [2][11][25][30], such as fractional Brownian motion, and non-Gaussian self-similar processes, such as $\alpha-$stable self-similar process [3][24], and the $(\sigma(\theta),\rho(\theta))$ stochastic traffic model [7][9]. With Theorem 3, the following example shows that gSBB can be readily represented using the time-domain $v.w.d$ SAC.

If an arrival process $\mathcal{A}(t)$ can be described by gSBB with upper rate $\rho$ and bounding function $f(x)\in\bar{\mathcal{F}}$, i.e., for any $t,x\geq 0$, there holds

then the process $\mathcal{A}(t)$ has a $v.b.c$ SAC $\alpha(t)=\rho\cdot t$ with the bounding function $f(x)$. With Theorem 3 (1), the arrival process has a $v.w.d$ SAC $\lambda(n)=\frac{n}{\rho}$ which is sub-additive and the bounding function $h(y)=f(\rho\cdot y)$, i.e.,

Remark. Theorem 3 allows us to readily utilize the results of gSBB traffic for time-domain models. If the traffic is more suitable for being characterized by the time-domain traffic models rather than the space-domain traffic models, then the transformation between two domains can facilitate the analysis.

Queueing theory characterizes the service process of a system based on the per customer service time. Like the arrival model, time-domain service models extend to the cumulative service time.

If packet $P(n)$ arrives to a network system after packet $P(n-1)$ has departed from the system, the departure time of $P(n)$ is the arrival time $a(n)$ plus the service time $\delta_{n}$, i.e., $a(n)+\delta_{n}$. If $P(n)$ arrives to the system while $P(n-1)$ is still in the system, then its departure time is $d(n-1)+\delta_{n}$. The combination of both cases gives the departure time of $P(n)$

with $d(0)=a(0)+\delta_{0}$. Applying Eq.(12) iteratively to its right-hand side results in

The system usually allocates a minimum service rate to an arrival flow in order to meet its QoS requirements. The guaranteed minimum service rate is related to the guaranteed maximum service time for each packet of the flow. Accordingly, the time that the packet departs from the system is bounded. Denote the guaranteed maximum service time by $\hat{\delta}_{n}$. The Guaranteed Rate Clock (GRC) is defined based on $\hat{\delta}_{n}$ [17] [18]:

with $GRC(0)=a(0)+\hat{\delta}_{0}$. Applying Eq.(14) iteratively to its right-hand side yields

Eq.(15) is similar to Eq.(13) except for that $GRC(n)$ represents the guaranteed departure time⁴⁴4The guaranteed departure time is actually $GRC(n)+$error term [17], where error term is determined by the employed service discipline. The underlying service discipline considered throughout this paper is FIFO, under which, the error term is $zero$. while $d(n)$ is the actual departure time.

If $\sum_{i=m}^{n}\hat{\delta}_{i}$ is denoted by a function $\gamma(n-m+1)$, then Eq.(15) becomes

which is the basis for the time-domain (deterministic) service model [10]. For systems that only provide service guarantees stochastically or applications that require only stochastic QoS guarantees, the service time may not need to be deterministically guaranteed. In this case, we extend the (deterministic) service curve into a probabilistic one.

Note that the stochastic service curve of a service process is not unique. Therefore optimization is needed to find the SSC of a specific system.

Consider two nodes, the transmitter and the receiver. They communicate through an error-prone wireless link which is modeled as a slotted system. The wireless link can be considered as a stochastic server. Packets have fixed-length and are served in a FIFO manner by the transmitter. To simplify the analysis, we assume that the length of time slot equals one packet transmission time⁵⁵5It means we only compute the number of time slots in this example..

The transmitter sends packets only at the beginning of a time slot. Due to the error-prone nature of the wireless link, the probability that a packet is successfully transmitted is determined by packet error rate (PER). Here, we assume that packet errors happen independently in every transmission with a fixed PER denoted by $P_{e}$. The successful transmission probability of one packet is hence $1-P_{e}$. If error happens, the unsuccessfully transmitted packet will be retransmitted in the next time slot immediately. In order to guarantee 100% reliability, the packet will be retransmitted until it is successfully received by the receiver.

The per-packet service time $\delta_{n}$ is a geometric random variable with parameter $1-P_{e}$. The cumulative service time of successfully transmitting packets $P(m)$ to $P(n)$ is $\sum_{k=m}^{n}\delta_{k}$ which follows the negative binomial distribution with parameter $1-P_{e}$. The mean service time denoted by $\bar{\delta}$ equals $\frac{1}{1-P_{e}}.$

According to the complementary cumulative distribution function (CCDF) of the negative binomial distribution, the cumulative service time between two arbitrary packets $P(m)$ and $P(m+n)$ is given by

for any $x\geq 0$, where $\lceil\cdot\rceil$ is the ceiling function.

The right-hand side of Eq.(18) represents the bound on the probability that the actual cumulative service time exceeds the cumulative mean service time. Let $\gamma_{\eta}(n)=\bar{\delta}\cdot n+\eta\cdot n$ for $\eta>0$ and $j(x)$ denote the right-hand side of Eq.(18). From Definition 7, we know

from which, we have

Thus, we conclude that this error-prone wireless link provides an $i.d$ SSC $\gamma_{\eta}(n)$ with the bounding function $j_{\eta}(x)$ for $\eta>0$, where

Since Eq.(18) is only relevant to the cumulative service time and does not involve the arrival process, it provides a method to find the $i.d$ SSC.

Remark. Example 4 demonstrates that we can obtain the $i.d$ SSC from analyzing per-packet service time. However, if applying the space-domain results to this case, we need an impairment process [22] to characterize the cumulative amount of service consumed by unsuccessful transmissions. In other words, we still need to compute the cumulative slots due to failed transmission and then convert it into the amount of service. Such conversion may introduce error or result in looser bounds whereas the time-domain model directly computes the service time and avoids the conversion error. This simple example thus illustrates the feasibility of the time-domain service curve model.

In Section 4, we show that many results can be derived from the $i.d$ SSC. However, without additional constraints, we have difficulty in proving the concatenation property for the $i.d$ SSC. To address this difficulty, we introduce another service curve model in the following.

Note that the left-hand side of Eq.(19) represents a property that is typically hard to calculate. It means that Definition 8 is more strict than Definition 7. Thus it is important to find the relationship between the $i.d$ SSC and the $\eta$-stochastic service curves.

Again, in the second part of Theorem 4, $j(x)\in\bar{\mathcal{G}}$ while not $j(x)\in\bar{\mathcal{F}}$. If the requirement on the bounding function is relaxed to $j(x)\in\bar{\mathcal{F}}$, the above relationship may not hold in general.

Definition 7 explores the relationship between the arrival process and the departure process, but it does not explicitly characterize the service process. From Eq.(17), it is not trivial to find the stochastic service curve $\gamma(n)$ for a specific system. Example 4 illustrates how to add some increment $\eta$ to the stochastic service curve $\gamma(n)$. To this end, we expand Eq.(16) to

Without loss of generality, assume $a(m_{0})$ ($0\leq m_{0}\leq n$) is the beginning of the backlogged period in which packet $P(n)$ is served. Then,

and $a\bar{\otimes}\gamma(n)\geq a(m_{0})+\gamma(n-m_{0}+1)$.

We rewrite the right-hand side of Eq.(20) as

Note that Eq.(21) holds for arbitrary $m_{0}\leq n$. Inspired by this, we define a new service curve model.

Eq.(21) reveals a relationship between the $i.d$ SSC and the stochastic strict service curve. Furthermore, in Theorem 4(2), the relationship between the stochastic strict service curve and the $\eta-$stochastic service curve is obtained.

Note that the second part of Theorem 5 requires the bounding function $j(x)\in\bar{\mathcal{G}}$ while not $j(x)\in\bar{\mathcal{F}}$.

Suppose that the arrival process has a $v.w.d$ SAC and the service process has an $i.d$ SSC. Under this condition, we derive the delay bound and backlog bound.