New CRT sequence sets for a collision channel without feedback
Protocol sequences are binary and periodic sequences used for deterministic multiple access in a collision channel without feedback. In this paper, we focus on user-irrepressible (UI) protocol sequences that can guarantee a positive individual throughput per sequence period with probability one for a slot-synchronous channel, regardless of the delay offsets among the users. As the sequence period has a fundamental impact on the worst-case channel access delay, a common objective of designing UI sequences is to make the sequence period as short as possible. Consider a communication channel that is shared by active users, and assume that each protocol sequence has a constant Hamming weight . To attain a better delay performance than previously known UI sequences, this paper presents a CRTm construction of UI sequences with , which is a variation of the previously known CRT construction. For all non-prime , our construction produces the shortest known sequence period and the shortest known worst-case delay of UI sequences. Numerical results show that the new construction enjoys a better average delay performance than the optimal random access scheme and other constructions with the same sequence period, in a variety of traffic conditions. In addition, we derive an asymptotic lower bound on the minimum sequence period for if the sequence structure satisfies some technical conditions, called equi-difference, and prove the tightness of this lower bound by using the CRTm construction.
Keywords:collision channel without feedback protocol sequences user-irrepressible sequences CRT sequences conflict-avoiding codes
Protocol sequences are periodic deterministic binary sequences that are used to provide reliable medium access control (MAC) protocol for a collision channel without feedback Massey85 (). Compared with time division multiple access (TDMA), ALOHA and carrier sense multiple access (CSMA), a protocol sequence-based scheme does not require stringent time synchronization, channel monitoring, backoff algorithm or packet retransmission. Such simplicity is particularly desirable in wireless sensor networks (WSNs) and vehicular ad hoc networks (VANETs), where well-coordinated transmission and time synchronization may be difficult to achieve due to user mobility, time-varying propagation delays or energy constraints. The natural interest of protocol sequence based schemes, the guaranteed performance metrics, such as worst-case delay or minimum throughput, have been commonly considered in previous studies on protocol sequences Wong07 (); CWS08 (); SCSW (); VANET13 (); VANET14 (); Wong14 (); MS16 (); XCW16 (); CYCLWK16 (); ZLWS16 (); SCCW16 (). In addition, some further performance metrics, such as average group/individual delay or average throughput, have been investigated in SCSW (); ZLWS16 (); SCCW16 (), and related approaches for sequence allocation can be found in VANET14 (); Wong14 (); MS16 (); XCW16 ().
In this paper, our focus is on user-irrepressible (UI) protocol sequences that can guarantee a positive individual throughput per sequence period with probability one for a slot-synchronous channel. UI property is a fundamental requirement in a protocol sequence-based scheme for delay-constrained services with small amounts of user data. The design goal of UI sequences is to minimize the sequence period, which indicates how long the receiver has to wait between two successfully transmitted packets in the worst-case, when the number of active users are given. Some constructions of UI sequences can be found in Massey85 (); Wong07 (); CWS08 (); SCSW (); CYCLWK16 (); Nguyen92 (); GyorfiVajda93 (); BE95 (); YK95 (); YK02 (); SWSC (); CRT (); UIS09 ().
We remark that deterministic MAC protocols can also be referred to in the literature as conflict-avoiding codes (CACs)LT05 (); CAC32010 (); SWC10 (); LFL15 (); LMJ16 (), optical orthogonal codes (OOCs)CSW89 (); WJ10 (); FWWZ16 (), or topological transparent scheduling CF94 (); CCS06 (); CCS06-2 (); LLLZ14 () with different design goals. In particular, UI sequences aim to minimize the sequence period by assuming all users are active, whereas CACs aim to maximize the number of potential users when the sequence period and the number of maximum active users are both given. In addition to the aforementioned deterministic access schemes, other combinatorial designs for applications in communications, cryptography, and networking can be found in CDSLP99 ().
1.2 System Model
We consider a feedback-free multiple-access channel shared by users, all of them may be active simultaneously, transmitting to a single receiver. This model is applicable to bursty traffic. Channel time is assumed to be divided into time slots of equal duration. Each active user reads out the sequence entry from the assigned binary sequence sequentially and transmits a packet within a time slot if and only if the sequence value is equal to 1.
To model a time slotted system, time indices are in units of one time slot duration. Due to propagation delay, user mobility or random traffic, there are relative time offsets of user for , such that a packet from user , received at the time instant on the receiver’s clock, was actually sent at the time instant on user ’s clock. These relative time offsets are random, always unknown to the users, but unchanged in a communication session. As introduced in Massey85 (); CCS06 (), there are two different levels of channel synchronization:
The channel is slot-synchronous if is an arbitrary integer for all , i.e., all users know the slot boundaries and transmit packets aligned to the slot boundaries.
The channel is completely asynchronous if is an arbitrary real number for all .
For practical considerations, the slot-synchronous assumption is valid if the synchronization is provided by simple narrow band beacon signals. In these scenarios, the receiver can only approximate by (modulo ) for all , where is the used frequency of the beacon signal. As such, for simplicity, we restrict our attention to the slot-synchronized model.
If exactly one user transmits a packet within a slot, the packet can be received correctly. A collision occurs if two or more than two users transmit simultaneously, and all time-overlapping packets are assumed unrecoverable.
For , let be a binary protocol sequence with sequence period (or length) assigned to user . Let denote the ring of residues modulo . Given , we define the characteristic set of by . We also call as a “sequence”, although it is actually represented as a subset of . The cardinality of , , is called the Hamming weight of or . Let
where the addition is performed in , be the shifted version of by a relative shift . If the relative time offset of user is , he or she transmits a packet at time slot if and only if in modulo . This paper assumes all sequences have the same Hamming weight (called constant-weight) and share the same period .
For the sake of convenience, for any positive integer , let and . Obviously, whenever . Let be a collection of subsets in . is said to be unblocked in if for any integer pattern , one has
We say is user-irrepressible (UI) if is unblocked for all .
One can check that the following is a UI sequence set of period by (1).
Given two characteristic sets , and a relative shift between them. Let denote the Hamming cross-correlation between and with respect to by giving
The maximum Hamming cross-correlation between and is defined as
By symmetry, one can see that . Let be the maximum Hamming cross-correlation for any pair of distinct characteristic sets in . We note that measures the maximal mutual interference between any pair of the users in a slot-synchronous channel. In Example 1, one can check that , and . Hence we have .
We remark here that for practical considerations, one would like to remove the slot-synchronous assumption. It is, in fact, possible to do so and to allow the users to be completely asynchronous. In a completely asynchronous channel Hui84 (), compared with the slot-synchronous one, the relative time offsets are arbitrary real numbers in units of one time slot duration. In such a channel, a collision can occur due to partial overlapping of packets. In other words, a packet is assumed to be successfully sent out if and only if it is not completely or partially overlapped by any other packet. A sequence set with a common period is said to be completely irrepressible (CI) ZSW11 () if each user can successfully send out at least one packet per period, no matter what the real offsets are. By definition, a CI sequence set must be UI. Conversely, a UI sequence set can be easily modified to provide the CI property at the cost of doubling the sequence period, by padding an extra zero after each sequence entry Massey85 (); CCS06 (); ZSW11 (). This approach ensures that the maximal number of conflicts occurring in any group of users remains unchanged when the system synchronism is reduced from the slot-synchronous to completely asynchronous. As such, we focus on the construction of UI sequences in this paper, and the generalization to the CI sequences is not considered.
1.3 Related Works and Motivation
There are various known UI sequences in literature: Shift-invariant Sequences (SI)Massey85 (); CWS08 (); SCSW (), Wobbling Sequences (WS)Wong07 (), Extended Prime Sequences (EPS)YK95 (), the Chinese Reminder Theorem Sequences (CRT)SWSC (); CRT () and CRT Sequences in Prime Users (CRTp)UIS09 (). Table 1 lists the major parameters of these UI sequences.
|SI Massey85 (); CWS08 (); SCSW ()||positive integer||Yes|
|WS Wong07 ()||odd prime||Yes|
|EPS YK95 ()||odd prime||Yes|
|CRT SWSC (); CRT ()||positive integer||Yes|
|CRTp UIS09 ()||odd prime||No|
The primary design objective of UI sequences is to minimize the sequence period when is given, as has a fundamental impact on the worst-case channel access delay, i.e., the maximum waiting time that a message can be successfully received. Let be the smallest such that a set of UI sequences of common period exists. The work in UIS09 () shows that is lower bounded by . One can see from Table 1 that SI sequences are the shortest known UI sequences for , CRTp sequences are the shortest for all prime , and CRT sequences are the shortest for all non-prime . Moreover, the work in SWSC () improves the lower bound on the sequence period from to for the case with constant Hamming weight , and shows that the CRT sequences achieve this lower bound asymptotically.
We now know that CRT sequences are the shortest known UI sequences for all non-prime , however, their small Hamming weight would possibly yield larger average delay VANET13 (); VANET14 (), which is also an important considered metric in the evaluation of channel access schemes. This observation motivates us to investigate short UI sequences with . In this paper, we propose a CRTm construction of UI sequences, which is also based on the Chinese Remainder Theorem Ireland90 ().
Our proposed CRTm sequences are with period , constant Hamming weight and , where is the smallest prime that is larger than , for any positive integer . Obviously, when is a non-prime. Both CRT sequences and CRTm sequences are the shortest known UI sequences for all non-prime , but CRTm sequences have a larger Hamming weight. It will be shown that this larger Hamming weight would bring better average individual-delay and group-delay. In addition, similar to the improvement of minimum sequence period for in SWSC (), we derive an asymptotic lower bound of for if the sequence structure satisfies some technical conditions, called equi-difference, and hence prove the CRTm construction is optimal in the sense that it can achieve this lower bound. Our method can be viewed as a generalization of that in SWSC () for .
It is worth pointing out that our proposed CRTm construction allows , whereas all other known constant-weight UI sequences except SI in Table 1 strictly require . This latter condition clearly implies the UI property. The difference on the relation between and makes the proof for the UI property of the CRTm construction very different and more complicated.
The rest of this paper is organized as follows. After setting up some definitions and notation in Section 2, we provide the CRTm construction of UI sequences with constant weight in Section 3. Then, in Section 4, we establishe an asymptotic lower bound on the minimum sequence period with , and prove that the CRTm construction can achieve this lower bound if the sequences are constant-weight and equi-difference. Section 5 is devoted to demonstrate the delay performance of CRTm sequences through numerical study. A conclusion is given in Section 6.
2 Definitions and Notation
Given a sequence set . For , let be a collection of all indices () such that the , that is,
Let be a collection of all relative shifts such that
For in , we define
In Example 1, since , and , we have , and . Meanwhile, , , and .
Let be a sequence of period and Hamming weight . We use to denote the set of non-zero differences between pairs of distinct elements in , i.e.,
Obviously, . is said to be exceptional if . is called equi-difference if the elements in form an arithmetic progression in , i.e.,
where the product is performed in . The element is called a generator of . If all s in are equi-difference, then we say is equi-difference.
In Example 1, , , and are equi-difference with Hamming weight and generators , , and , respectively. One can further check that and thus is exceptional.
3 A New Construction of UI Sequences
We start this section with a necessary and sufficient condition for a sequence set to be UI.
Let be a sequence set. is UI if and only if, for , and arbitrary relative shifts , , one has
We only consider the necessary part as the sufficient part is simply obtained by letting . Assume to the contradiction that there exist , and , such that the cardinality of is at most . Then we always can choose some relative shifts for all such that
Now, we present a new construction of constant-weight UI sequences, called the CRTm construction, which is a variation of the CRT construction SWSC ().
Let and be relatively prime integers. Let , consisting of all ordered pairs with and , be the direct product of the rings and . There is a natural bijection (ring isomorphism) by defining
that is, preserves addition and multiplication on both sides. If there is no danger of confusion, operations under are componentwise taken modulo and . We will construct sequences by specifying characteristic sets in .
CRTm Construction: Given , we set to be the smallest prime larger than . Let
Notice that is relatively prime to due to and the Bertrand’s postulate, which states that there is always a prime strictly between and for any integer . Then we obtain the characteristic sets of the sequences, , by taking the inverse image for . The CRTm construction produces sequences of length and constant-weight .
Given , the CRTm construction produces 8 sequences of period 77 and constant-weight 7. The characteristic sets are:
The previously known CRT construction can be obtained from the CRTm construction by removing the last ordered pair from each sequence, namely, by letting . As the CRT construction allows SWSC (); CRT (), the UI property can be derived directly by . However, one can see from Example 2 that for CRTm construction. Even though the two constructions look similar, the proof of the UI property is very different as the CRTm construction allows .
Following Example 2, we consider the set , and try to show that is unblocked in . Observe that , , and . It is easy to see that , that is, only the relative shift satisfies . Let . One has , which implies that at most packets in can be blocked by and simultaneously. Since at most one packet in can be blocked by each of the other three sequences, is unblocked in .
The above example illustrates the idea of how to prove a set of sequences having to be UI. As for the CRTm construction, we aim to show that any sequences obtained by the construction form a UI sequence set. We start with the following lemma that gives an equivalent condition for the existence of UI sequences with constant-weight .
A sequence set of constant-weight is UI if and only if, for , one has
for any ; and
, i.e., , for any distinct integers such that and any .
First, we prove the necessary part by contradiction.
(i) Suppose for some . Then for some , which contradicts to (2) by setting .
(ii) Suppose for some distinct integers such that and some . By the defining property of , we have . It is easy to see . Since , it further implies that . Let be the relative shift so that . Then,
which contradicts to (2) by setting .
For the sufficient part, with condition (i) and (ii), as , it is easy to see that for any and any relative shifts . It implies that is UI. ∎
We are ready to prove the UI property of the CRTm construction.
Any sequences from the CRTm construction form an equi-difference UI sequence set of length and constant-weight .
Observe that has the generator for and generator when . Thus, the CRTm construction produces equi-difference sequences of length . We define in the same way as , but with the addition and subtraction done in instead of . It is sufficient to show that each obtained sequence , , satisfies the two conditions in Lemma 2. Note that, if for any , then both the two conditions of Lemma 2 hold for .
First, consider . We claim that . Suppose to the contradiction that , that is, , for some . Then
for with and . By equating the second components on both sides, we have , a contradiction to . Hence possesses the two conditions in Lemma 2.
Second, consider . If for any , then we are done; otherwise, for some , i.e., . Note that is not a candidate for due to the first part. implies that
for some with and . By equating the second components on both sides, we have (mod ). Since , there are only five possible solutions to and , as follows.
If , from the first component, we have (mod ), which implies that or due to and . This contradicts the assumption that and . Hence (4) can be excluded. The remaining four possible solutions (5)–(8) imply respectively (9)–(12).
Intuitively speaking, from the construction of
we call elements , , and the head, second-head, second-tail and tail, respectively. The above arguments conclude the following property, say collided property.
If there is a pair of repeated elements between two sequences under some relative shift, the two elements must be of one sequence, and or of another.
Back to , the collided property immediately implies for any , i.e., Lemma 2(i). Now, it remains to show that Lemma 2(ii) holds as well. Consider . If , then the result follows. If and , by the definition of and condition (i), we have , and . By the collided property, both the repeated elements of and are in the form , while the repeated elements of with respect to and are in the form (or ) and (or ). Since in a single sequence is a shifted version of , the two pairs of and are repeated under some relative shift, which contradicts to the collided property. Hence we complete the proof. ∎
4 A Tight Asymptotic Lower Bound on Sequence Period
In this section, we derive an asymptotic lower bound on sequence period for the equi-difference structure and , and then show this lower bound is tight by using the CRTm construction.
We start with three basic results which will be useful to derive a lower bound on sequence period.
The following necessary condition for of constant-weight to be UI directly follows from Lemma 2(ii), and thus its proof is omitted here.
Let be a UI sequence set of constant-weight . Then for , one has
for , , , and .
The following result provides an upper bound on for .
Let and be two equi-difference sequences of the same period. If , then
Let and are the generators of and , respectively. Then and are of the form:
Consider the following four sets:
Obviously, and .
Now we prove this lemma by contradiction. Suppose . Then we have
which implies that at least one of the four sets , , and has cardinality at least 2. There are four cases as follows.
If , then .
If , then .
If , then .
If , then .
Each of the four cases implies that , which contradicts to the assumption that