Stochastic Service Guarantee Analysis Based on Time-Domain Models
Stochastic network calculus is a theory for stochastic service guarantee analysis of computer communication networks. In the current stochastic network calculus literature, its traffic and server models are typically defined based on the cumulative amount of traffic and cumulative amount of service respectively. However, there are network scenarios where the applicability of such models is limited, and hence new ways of modeling traffic and service are needed to address this limitation. This paper presents time-domain models and results for stochastic network calculus. Particularly, we define traffic models, which are defined based on probabilistic lower-bounds on cumulative packet inter-arrival time, and server models, which are defined based on probabilistic upper-bounds on cumulative packet service time. In addition, examples demonstrating the use of the proposed time-domain models are provided. On the basis of the proposed models, the five basic properties of stochastic network calculus are also proved, which implies broad applicability of the proposed time-domain approach.
Stochastic network calculus is a theory dealing with queueing systems found in computer communication networks 
. It is particularly useful for analyzing networks where service guarantees are provided stochastically. Such networks include wireless networks, multi-access networks and multimedia networks where applications can tolerate some certain violation of the desired performance .
Stochastic network calculus is based on properly defined traffic models  and server models . In the existing models of stochastic network calculus, an arrival process and a service process are typically modeled by some stochastic arrival curve, which probabilistically upper-bounds the cumulative amount of arrival, and respectively by some stochastic service curve, which probabilistically lower-bounds the cumulative amount of service. In this paper, we call such models space-domain models. Based on the space-domain traffic and server models, a lot of results have been derived for stochastic network calculus. Among the others, the most fundamental ones are the five basic properties  : (P.1) Service Guarantees including delay bound and backlog bound; (P.2) Output Characterization; (P.3) Concatenation Property; (P.4) Leftover Service; (P.5) Superposition Property. Examples demonstrating the necessity of having these basic properties and their use can be found  .
Nevertheless, there are still many open research challenges for stochastic network calculus, and a critical one is time-domain modeling and analysis . Time-domain modeling for service guarantee analysis has its root from the deterministic Guaranteed Rate (GR) server model , where service guarantee is captured by comparing with a (deterministic) virtual time function in the time-domain. This time-domain model has been extended to design aggregate-scheduling networks to support per-flow (deterministic) service guarantees , while few such results are available from space-domain models. Other network scenarios where time-domain modeling may be preferable include wireless networks and multi-access networks.
In wireless networks, the varying link condition may cause failed transmission when the link is in ‘bad’ condition. The sender may hold until the link condition becomes ‘good’ or re-transmit. For such cases, it is difficult to directly find the stochastic service curve in the space-domain because we need to characterize the stochastic nature of the impaired service caused by the ‘bad’ link condition. A possible way is that we use an impairment process  to characterize the impaired service. However, how to define and find the impairment process arises another difficulty. Even though we can define an impairment process, we may first convert the impairment process into some existing stochastic network calculus models, and then further analyze the performance bounds. The obtained performance bounds may become loose because of such conversion. If we characterize the serivce process in the time-domain, we can use random variables to represent the time intervals when the link is in ‘bad’ condition. Analyzing the stochastic nature of such random variables would be easier. In addition, this way can avoid the difference introduced by the intermediate conversion.
In contention-based multi-access networks, backoff schemes are often employed to reduce collision occuring. Because the backoff process is characterized by backoff windows which may vary with the different backoff stages, it is quite cumbersome for a space-domain server model to characterize the service process with the consideration of the backoff process. This also prompts the possibility of characterzing the service process in the time-domain. Having said this, however, how to define a stochastic version of the virtual time function and how to perform the corresponding analysis are yet open .
The objective of this paper is to define traffic models and server models in the time-domain and derive the corresponding five basic properties for stochastic network calculus. Particularly, we define traffic models that are based on probabilistic lower bounds on cumulative packet inter-arrival time. Also, we define server models that are based on some virtual time function and probabilistic upper bounds on cumulative packet service time. In addition, we establish relationships among the proposed time-domain models, and the mappings between the proposed time-domain models and the existing space-domain models. Furthermore, we prove the five basic properties based on the proposed time-domain models.
The remainder is structured as follows. Sec. II introduces the mathematical background and fundamental space-domain models and relevant results of stochastic network calculus. In Sec. III, we first introduce the time-domain deterministic traffic and server models, and then extend them to stochastic versions. In addition, the relationships among them as well as with some existing space-domain models are established. Sec. IV explores the five basic properties. Sec. V summarizes the work.
Ii Notation and Background
To ease expression, we assume networks with fixed unit length111The results can also be extended to networks with variable-length packets while the expression and results will be more complicated. packets. By convention, we assume that a packet is considered to be received by a network element when and only when its last bit has arrived to the network element, and a packet is considered out of a network element when and only when its last bit has been transmitted by the network element. A packet can be served only when its last bit has arrived. All queues are assumed to be empty at time . Packets within a flow are served in the first-in-first-out (FIFO) order.
Let , , and denote the packet of a flow, its allocated service rate, its arrival time and its departure time, respectively. Let and respectively denote the number of cumulative arrival packets and the number of cumulative departure packets by time . By convention, we assume , , and . For any , we denote and .
In this paper, and will be used to represent an arrival process interchangeably. A departure process will be represented by and interchangeably.
The following function sets are often used in this paper. Specifically, we use to denote the set of non-negative wide-sense increasing functions as follows:
We denote by the set of non-negative wide-sense decreasing functions:
Let denote the set of functions in , where for each function , its nth-fold integration, denoted by , is bounded for and still belongs to for , or
For ease of exposition, we adopt
and assume that for any bounding function , for .
Ii-B Max-plus and Min-plus Algebra Basics
An essential idea of (stochastic) network calculus is to use alternate algebras particularly the min-plus algebra and max-plus algebra  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 are mainly under the space-domain and based on min-plus algebra that has basic operations particularly suitable for characterizing cumulative arrival and cumulative service. For characterizing arrival and service processes in the time-domain, interestingly, the max-plus algebra has basic operations that well suit the need.
In this paper, the following max-plus and min-plus operations are often used:
Max-Plus Convolution of and is
Max-Plus Deconvolution of and is
Min-Plus Convolution of and is
Min-Plus Deconvolution of and is
In this paper, when applying supremum and infimum, they may be interpreted as maximum and minimum whenever appropriate, respectively.
The following lemma is often used for later analysis and thus listed:
For the sum of a collection of random variables , no matter whether they are independent or not, there holds for the complementary cumulative distribution function (CCDF) of : (See Lemma 1.5 in )
where , .
For later analysis, we need some transformation between the number of cumulative arrival packets by time , i.e., , and the time of a packet arriving to the system, i.e., .
If is upper-bounded with respect to some function , we have the following lemma.
For function , there holds:
the following statements are equivalent:
, for ;
, for ;
if , holds, then we have , where is the inverse function of and defined as follows
(1) For , from the condition, we obtain for . Then, there holds
Thus, we conclude for .
For , from the condition, we have
Then there must be for . Thus, holds for and .
Example 1. Suppose the number of cumulative arrival packets of a flow, , is upper-bounded by for , where . Let . We can get the inverse function of , . We can use Eq.(3) to get . For , we have
from which we get . Then, we know that for any packet, its arrival time satisfies
If is lower-bounded with respect to some function , we have the following lemma.
For function , there holds:
the following statements are equivalent:
, for ;
, for ;
if , holds, then we have , where is the inverse function of and defined as follows
(1) The part has been proved in Lemma 2(2). We only prove the part. From the condition, we have
Thus there holds for and .
(2) For , we can find according to the following functions
Thus, we have and . From (1), we know that is equivalent to . Then we have
Taking the inverse function of yields
Because , we have
Let . The above inequality is written as
Since holds for , we have
from which we further obtain
We then conclude . ∎
Ii-D Related Space-domain Results
This sub-section reviews some related space-domain results under min-plus algebra . It is worth highlighting that the following results are for discrete time systems with unit discretization step.
The virtual-backlog-centric (v.b.c) stochastic arrival curve model  is defined based on a probabilistic upper-bound on cumulative arrival. To ease later analysis, the definition of v.b.c stochastic arrival curve model presented in this paper is based on the number of cumulative arrival packets while not the amount of cumulative arrival (in bits) which has been widely used in the network calculus literature.
The v.b.c stochastic arrival curve model explores the virtual backlog property of deterministic arrival curve, which is that the queue length of a virtual single server queue (SSQ) fed with the same flow with a deterministic arrival curve is upper-bounded.
For a flow having arrival curve , we construct a virtual SSQ system fed with the same flow. The SSQ system has infinite buffer space and the buffer is initially empty. Suppose the virtual SSQ system provides service to the flow for all . Then the unfinished work or backlog in the virtual SSQ system by time is . The Lindely equation can be used to derive , which is
Eq.(8) means that the amount of traffic backlogged in the system by time equals the amount of traffic backlogged by time plus the amount of traffic having arrived between and minus the amount of traffic having been served between and . By applying Eq.(8) iteratively to its right-hand side, it becomes
If the flow is constrained by arrival curve for all , it follows from Eq.(9) that the system backlog is also upper-bounded by . The v.b.c stochastic arrival curve is defined based on the virtual backlog property.
(v.b.c Stochastic Arrival Curve).
A flow is said to have a virtual-backlog-centric (v.b.c) sto-
chastic arrival curve with bounding function , denoted by , if for all and all , there holds
Eq.(10) can also be written as follows:
Based on the existing space-domain traffic and server models, a lot of results have been derived for stochastic network calculus which include the five basic properties  as introduced in Sec. I. In this paper, the following result is specifically made use of in later analysis and hence listed:
(Superposition Property). Consider flows with arrival processes , i=1,…,N, respectively. Let denote the aggregate arrival process. If , , then with , and .
Iii Time-Domain Models
This section reviews the deterministic arrival curve and the deterministic service curve models defined in the time-domain. We generalize the deterministic models and define time-domain stochastic arrival curve and stochastic service curve models.
Iii-a Deterministic Arrival Curve
Consider a flow of which packets arrive to a system at time . In order to deterministically guarantee a certain level of quality of service (QoS) to this flow, the traffic sent by this flow must be constrained. The deterministic network calculus traffic model in the time-domain characterizes packet inter-arrival time using a lower-bound function, called arrival curve in this paper and defined as follows :
(Arrival Curve). A flow is said to have a (deterministic) arrival curve , if its arrival process satisfies, for all ,
The arrival curve model has the following triplicity principle which will be used as the basis in defining the stochastic arrival curve models in the subsequent subsections.
The following statements are equivalent:
, for ;
, for ;
, for ,
It is trivially true that
from which, (2) implies (1). In addition
with which, (3) implies (2).
For (1)(2), it holds since for . For (2)(3),
Thus (1), (2) and (3) are equivalent. ∎
From Definition 2, the right-hand side of in Lemma 5.(1) defines an arrival curve . In addition, we can construct a virtual single server queue (SSQ) system that is initially empty, fed with the same traffic flow, and has a service curve which makes (see Definition 6). Then, the delay in the virtual SSQ system is upper-bounded by , and the maximum system delay for the first packets is upper-bounded by
Example 2. The Generic Cell Rate Algorithm (GCRA)  with parameter is a parallel algorithm to the Leaky Bucket algorithm and has been used in fixed-length packet networks such as Asynchronous Transfer Mode (ATM) networks. The GCRA measures cell rate at a specified time scale and assumes that cells will have a minimum interval between them. Here, denotes the assumed minimum interval between cells and denotes the maximum acceptable excursion that quantifies how early cells may arrive with respect to . It can be verified that if a flow is GCRA-constrained, it has an arrival curve
Iii-B Inter-arrival-time Stochastic Arrival Curve
Lemma 5.(1) defines a deterministic arrival curve which lower-bounds the inter-arrival time between any two packets. Based on this, we define its probabilistic counterpart as follows:
(i.a.t Stochastic Arrival Curve). A flow is said to have an inter-arrival-time (i.a.t) stochastic arrival curve with bounding function , denoted by , if for all and all , there holds
Example 3. Consider a flow with fixed unit packet size. Suppose its packet inter-arrival times follow an exponential distribution with mean . Then, the packet arrival time has an Erlang distribution with parameter . And, for any two packets and , their inter-arrival time satisfies, for ,
The i.a.t stochastic arrival curve is intuitively simple, but it has limited use if no additional constraint is enforced. Let us consider a simple example to understand this problem. Consider a single node with constant per packet service time and its input flow satisfying where . Suppose we are interested in the delay , where, by definition, . Because the node has constant per packet service time , it has a (deterministic) service curve which implies . Then we have
From Eq.(14), we have difficulty in further deriving more results if no additional constraint is added because we only know . When investigating the performance metrics such as delay bound and backlog bound in Section IV-A, we meet the similar difficulty.
Iii-C Virtual-system-delay Stochastic Arrival Curve
The previous subsection stated the difficulty of applying i.a.t stochastic arrival curve to service guarantee analysis. This subsection introduces another stochastic arrival curve model that can help avoid such difficulty. This model is called virtual-system-delay () stochastic arrival curve. The model explores the virtual system delay property of deterministic arrival curve as implied by Lemma 5.(2), which is that the amount of time a packet spends in a virtual SSQ fed with the same flow with a deterministic arrival curve is lower-bounded.
For a flow having deterministic arrival curve, we construct a virtual SSQ system fed with the flow, which has infinite buffer space and the buffer is initially empty. Suppose the virtual SSQ system provides a deterministic service curve to the flow or for all . The amount of time packet spends in the virtual SSQ system is = . If the flow is constrained by arrival curve for all , is also lower-bounded by .
Based on the virtual system time property, we define virtual-system-delay (v.s.d) stochastic arrival curve to characterize the arrival process.
(v.s.d Stochastic Arrival Curve). A flow is said to have a virtual-system-delay (v.s.d) stochastic arrival curve with bounding function , denoted by , if for all and all , there holds
Eq.(15) can also be written as
can be considered as the expected time that the packet would arrive to the system if the flow had passed through the virtual SSQ with service curve . denotes the difference between the expected arrival time and the actual arrival time. Eq.(16) characterizes this difference by introducing a bounding function .
Example 4. Consider a flow with the same fixed packet size. Suppose all packet inter-arrival times are exponentially distributed with mean . Based on the steady-state probability mass function (PMF) of the queue-waiting time for an M/D/1 queue , we say that the flow has a v.s.d stochastic arrival curve for , with and
where, denotes the greatest integer less than or equal to .
The following theorem establishes relationships between i.a.t stochastic arrival curve and v.s.d stochastic arrival curve.
If a flow has a v.s.d stochastic arrival curve with bounding function , then the flow has an i.a.t stochastic arrival curve with the same bounding function .
Inversely, if a flow has an i.a.t stochastic arrival curve with bounding function , it also has a v.s.d stochastic arrival curve with bounding function where
The first part follows from that
holds for . For the second part, there holds
Since for ,
Note that in the second part of the above theorem, while not . If the requirement on the bounding function is relaxed to , the above relationship may not hold in general.
The v.s.d stochastic arrival curve has a counterpart defined in the space-domain, the v.b.c stochastic arrival curve as defined in Definition 1. The following theorem establishes relationships between these two models.
If a flow has a v.b.c stochastic arrival curve with bounding function , the flow has a v.s.d stochastic arrival curve with bounding function , where and .
If a flow has a v.s.d stochastic arrival curve with bounding function , the flow has a v.b.c stochastic arrival curve with bounding function , where and .
We further have
From the condition that the flow has a v.b.c stochastic arrival curve , we know . According to Eq.(7), we obtain
Iii-D Maximum-(virtual)-system-delay Stochastic Arrival Curve
The maximum-(virtual)-system-delay (m.s.d) stochastic arrival curve explores the maximum virtual system delay property of deterministic arrival curve implied by Lemma 5.(3), which is that the maximum system delay of a virtual SSQ fed with the same flow with a deterministic arrival curve is lower-bounded.
Similar to the discussion for v.s.d stochastic arrival curve, for a flow having arrival curve, we construct a virtual SSQ system fed with the flow, which has infinite buffer space and the buffer is initially empty. Suppose the virtual SSQ system provides a deterministic service curve to the flow or for all . The maximum system delay in the virtual SSQ system for the first arrival packets as . If the flow is constrained by arrival curve for all , the maximum system delay in the virtual SSQ is also upper-bounded by .
Based on the maximum virtual system delay property, we define m.s.d stochastic arrival curve model.
(m.s.d Stochastic Arrival Curve).
A flow is said to have a maximum-(virtual)-system-delay (m.s.d) stochastic arrival curve with bounding function , denoted by , if for all and all , there holds
Iii-E Deterministic Service Curve
To provide service guarantees to an arrival-constrained flow , the system usually needs to allocate a minimum service rate to . A guaranteed minimum service rate is equivalent to a guaranteed maximum service time for each packet of the flow, and accordingly the packet’s departure time from the system is bounded. Because packets of the same flow are served in FIFO manner, any packet from this flow will depart by which is iteratively defined by
with , where is the service time guaranteed to . By applying Eq.(19) iteratively to its right-hand side, it becomes
where is the guaranteed cumulative service time for packet to . Suppose we can use a function to denote , i.e. . Then, Equation(20) becomes
which provides a basis for the following time-domain (deterministic) server model that charaterizes the service using an upper bound on the cumulative packet service time :
(Service Curve). Consider a system with input process and output process . The system is said to provide to the input a (deterministic) service curve , if for ,
The (deterministic) service curve model has the following duality principle:
For , for all , if and only if for , where