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Designing of fault-tolerant computer system structures using residue number systems

Designing of fault-tolerant computer system structures using residue number systems 1IntroductionCurrently, the main directions of computer systems and components (CSC) development in a positional number system (PNS) are improving the speed of implementation of arithmetic operations (including integer arithmetic operations) and reliability of their functioning [1,2,3].The complexity, scale and volume of various computational tasks and tasks of managing large and complex information systems which are solved by computer system (CS) necessitate the expansion of functions and capabilities of the CS. It leads to an increase in the number and complexity of computing equipment and system equipment, and in the complexity of their software [4,5]. This, in turn, necessitates the adoption of additional measures to ensure the high reliability of the CS and, first of all, the need to take additional measures to ensure the high fault tolerance of the CSC [6,7].2State-of-the-artThere are two main principles of increasing the reliability of the CS that operates in the PNS [8,9,10]: increasing reliability of individual elements of the CS (using a new base element) and introducing various types of redundancies. Since the reliability of logical elements is mainly determined by the current level of technology development, it is obvious that the introduction of redundancy when using any available element base is the most effective way to increase the reliability of the computer-controlled system [11,12].The listed measures for introducing redundancy ensure the availability of fault-tolerant properties in CS. The presence of fault-tolerance properties in the CS allows us to increase the reliability of its functioning. The fault-tolerant property [7] provides the ability to perform specified computational functions after failures, both by reducing, within acceptable limits, any indicators of the quality of functioning (e.g., by gradual degradation), and without deteriorating the quality of functioning of the computer-controlled system [2,13,14].The following is shown in the literature on this subject [2,15,16]. First, the use of binary PNS as the number system of the CS does not allow us to dramatically increase productivity and reliability. Second, there are results of basic research, which show possibility of a significant increase in the speed of implementation of integer arithmetic operations: addition, multiplication and subtraction by using a non-positional residue number system (RNS). This is achieved through the use of the properties of numbering systems, namely, independence, equal rights and low rate of transfer of residuals, which determine the non-positional code structure (NCS). The first and second properties of RNSs allow us to parallelize arithmetic calculations at the level of decomposition of residue numbers. Also, they provide an opportunity to realize spatial diversity of data elements with the possibility of their subsequent asynchronous independent processing. The use of the third properties of residue classes allow us to implement tabular methods for performing integer arithmetic operations of the basic set and implement polynomial functions with a single-cycle sample of the result of a modular operation. Thus, the use of RNS as number system of CS significantly (compared with PNS) increases the speed of the CSC.Suppose that at the design stage it is necessary to provide the needed (predetermined) level of fault-tolerance of a CS. It is possible to increase (ensure) the reliability if the system has a certain property, the use of which will allow it to be done [6,17]. Such property is defined and called fault-tolerance [7,18,19]. The presence of the fault tolerance property of the CS can be ensured both through the application of the method of passive fault tolerance (using the method of constant structural redundancy) [6,20] and through the method of active fault tolerance [21] (dynamic backup method) by introducing, in addition to the main computing paths CS in RNS, backup computing paths.In refs [14,22], it was noted that the use of passive fault-tolerant methods provides increased reliability of the CS in RNS in comparison with the majority turned computing system widely used in the PNSs. However, the lack of research results on the possibility of using residue classes to increase the reliability of the CS based on the application of active fault-tolerant methods limits the possibility of a comprehensive consideration of the task of ensuring the reliability of the functioning of the CSC. The purpose of the article is to design structures of a fault-tolerant CS in RNS in the case of applying the active fault-tolerant method.3Statement of the optimal redundancy problemBased on the properties of the RNS, the independence, equality and low number of digits of the residues representing the number of the structure of the CS in RNS are based on the use of the idea of structural redundancy in the PNS. In this case, the computing paths, according to the corresponding modules of the RNS, play the role of the main elements of the redundant system, and the control computing paths of the RNS play the role of the backup elements of the redundant system. Theoretically, the increase in the reliability of the CS in the RNS is explained as follows. First, based on the properties of independence and equality of the remainders, the totality of which determines the NCS, the structure of the CS in the RNS is a kind of a structural redundant system, consisting of computing paths according to the corresponding RNS modules. Each of the elements (computing paths) of this redundant system (CS) operates independently of each other and in parallel in time and each in its own separate module. This in the PNS is equivalent to the fact that smaller nodes of the CS are reserved. Second, in accordance with the provisions of the general reliability theory (from the point of view of increasing reliability), it is advisable to reserve small nodes and blocks of a complex system, since the failure rate of elements is always less than the failure rate of large nodes or the entire large system as a whole. The probability of fault-tolerant operation of the CS in the RNS is higher than the probability of fault-tolerant operation of the methods of increasing the reliability of duplicated and tripled majority computer structures widely used in practice. For the active fault-tolerant methods, the task of designing a fault-tolerant CS operating in the RNS is verbally formulated as follows: it is necessary to determine the optimal set {mj}OPT{\left\{{m}_{j}\right\}}_{{\rm{OPT}}}of modules from possible control mn+1,mn+2,…,mn+k{m}_{n+1},{m}_{n+2},\ldots ,{m}_{n+k}, at which reliability (fault tolerance) PRC(n/k)(t){P}_{{\rm{RC}}}^{(n\hspace{0.1em}\text{/}\hspace{0.1em}k)}\left(t)CS in the RNS will be maximum. However, there are limits on the amount of equipment VRC(n+k){V}_{{\rm{RC}}}^{\left(n+k)}CS in the RNS.To get the optimal set {mj}OPT{\left\{{m}_{j}\right\}}_{{\rm{OPT}}}of bases, it is necessary to formulate and solve the inverse problem of optimal reservation in RNS. If necessary, for the application of the active fault-tolerant method (application of the dynamic structural redundancy method), the inverse optimal reservation problem in RNS is mathematically formulated as follows (equation (1)): (1)PRC(n+z)(t)[t=const]→max;(VPNS(l)≥VRC(n+k)),\left\{\begin{array}{c}{P}_{{\rm{RC}}}^{\left(n+z)}\left(t){[}t={\rm{const}}]\to \max ;\\ ({V}_{{\rm{PNS}}}^{(l)}\ge {V}_{{\rm{RC}}}^{(n+k)}),\end{array}\right.where kkis a maximum possible number of control bases mn+1,mn+2,…,mn+k{m}_{n+1},{m}_{n+2},\ldots ,{m}_{n+k}(the number of CS control computing paths in residue classes); {m(n+z)}OPT(z=1…k){\{{m}_{\left(n+z)}\}}_{{\rm{OPT}}}\hspace{0.33em}\left(z=1\ldots k)is an optimal (providing a minimum amount of equipment VRC(n+k){V}_{{\rm{RC}}}^{(n+k)}CS in RNS) set of control bases (for a given value l of the CS character grid), the simultaneous use of which ensures the maximum fault-tolerance value PRC(n/z)(t){P}_{{\rm{RC}}}^{(n\hspace{0.1em}\text{/}\hspace{0.1em}z)}\left(t)CS in residue classes (RC).In this case, the following condition must be met: (2)VRC(k)≤VPNS(l)−VRC(n).{V}_{{\rm{RC}}}^{(k)}\le {V}_{{\rm{PNS}}}^{(l)}-{V}_{{\rm{RC}}}^{(n)}.It is convenient to solve the inverse optimal reservation problem in RNS with restriction by the method of coordinate wise steepest descent or by the method of dynamic programming [23].4Structures of CSs functioning in RNSLet us consider examples of constructing CS structures in RNS for l=1…4l=1\ldots 4. Tables 1 and 2 show the sets of information nnand control kkbases of the residue classes for the values of llof the corresponding characters grids l=1…4l=1\ldots 4of the CS. The data on the conditional relative amount of equipment of the computer station in RNS are presented, as well as the data on the conditional relative quantity of the equipment of CS in the PNS, reduced to one binary digit of the length llof the grid, where VRC(n)=∑i=1nαi,{V}_{{\rm{RC}}}^{(n)}=\mathop{\sum }\limits_{i=1}^{n}{\alpha }_{i},is a conditional relative amount of the equipment of the CS, reduced to one binary digit without excess CS in the RNS.VRC(k)=∑i=n+1n+kαi,{V}_{{\rm{RC}}}^{(k)}=\mathop{\sum }\limits_{i=n+1}^{n+k}{\alpha }_{i},is an additional conditional relative amount of CS equipment, reduced to one binary digit of CS in RNS, formed by the introduction of control bases. αi=[log2(mi−1)]+1{\alpha }_{i}={[}{\log }_{2}({m}_{i}-1)]+1is a number of bits needed to represent the mi{m}_{i}of the residue classes. ⌊X⌋\lfloor X\rfloor is an integer part of the number XXthat does not exceed it (rounding down). For example, [7.9] = 7. V0PNS(l){V}_{0{\rm{PNS}}}^{(l)}is a conditional relative amount of equipment without CS excess in the PNS. V2PNS(l){V}_{2{\rm{PNS}}}^{(l)}is a conditional relative amount of equipment of duplicated CS in the PNS. V3PNS(l){V}_{3{\rm{PNS}}}^{(l)}is a conditional relative amount of equipment of triple CS in the PNS.Table 1The sets of information nnand control kkbases of the residue classes, 2PNSl(n)l\left(n){mi},i=1…n\{{m}_{i}\},\hspace{0.33em}i=1\ldots nVRC(n){V}_{{\rm{RC}}}^{(n)}V2PNS(l){V}_{2{\rm{PNS}}}^{(l)}ΔV\Delta V{mj},j=n+1…n+k\{{m}_{j}\},\hspace{0.33em}j=n+1\ldots n+kVRC(k){V}_{{\rm{RC}}}^{(k)}kk1(4)m1=3,m2=4{m}_{1}=3,{m}_{2}=4,10166m5=11{m}_{5}=1141m3=5,m4=7{m}_{3}=5,{m}_{4}=72(6)2\left(6)m1=2,m2=5{m}_{1}=2,{m}_{2}=5,193213m7=17,m8=19{m}_{7}=17,{m}_{8}=19102m3=7,m4=9{m}_{3}=7,{m}_{4}=9,m5=11,m6=13{m}_{5}=11,{m}_{6}=133(8)m1=3,m2=4{m}_{1}=3,{m}_{2}=4,284820m9=23,m10=29,{m}_{9}=23,{m}_{10}=29,214m3=3,m4=7{m}_{3}=3,{m}_{4}=7,m11=31,m12=37{m}_{11}=31,{m}_{12}=37m5=11,m6=13{m}_{5}=11,{m}_{6}=13,m7=17,m8=19{m}_{7}=17,{m}_{8}=194(10)m1=2,m2=3{m}_{1}=2,{m}_{2}=3,376427m11=31,m12=37{m}_{11}=31,{m}_{12}=37,234m3=5,m4=7{m}_{3}=5,{m}_{4}=7,m13=41,m14=43{m}_{13}=41,{m}_{14}=43m5=11,m6=13{m}_{5}=11,{m}_{6}=13,m7=17,m8=19{m}_{7}=17,{m}_{8}=19,m9=23,m10=29{m}_{9}=23,{m}_{10}=29Table 2The sets of information nnand control kkbases of the residue classes, 3PNSl(n)l\left(n){mi},i=1…n\{{m}_{i}\},\hspace{0.33em}i=1\ldots nVRC(n){V}_{{\rm{RC}}}^{(n)}V2PNS(l){V}_{2{\rm{PNS}}}^{(l)}ΔV\Delta V{mj},j=n+1…n+k\{{m}_{j}\},\hspace{0.33em}j=n+1\ldots n+kVRC(k){V}_{{\rm{RC}}}^{(k)}kk1(4)m1=3,m2=4,{m}_{1}=3,{m}_{2}=4,102414m5=11,m6=13{m}_{5}=11,{m}_{6}=13,133m3=5,m4=7{m}_{3}=5,{m}_{4}=7m7=17{m}_{7}=172(6)m1=2,m2=5{m}_{1}=2,{m}_{2}=5,194829m7=17,m8=19{m}_{7}=17,{m}_{8}=19,255m3=7,m4=9{m}_{3}=7,{m}_{4}=9,m9=23,m10=29{m}_{9}=23,{m}_{10}=29,m5=11,m6=13{m}_{5}=11,{m}_{6}=13m11=31{m}_{11}=313(8)m1=3,m2=4{m}_{1}=3,{m}_{2}=4,287244m9=23,m10=29{m}_{9}=23,{m}_{10}=29,458m3=3,m4=7{m}_{3}=3,{m}_{4}=7,m11=31,m12=37{m}_{11}=31,{m}_{12}=37,m5=11,m6=13{m}_{5}=11,{m}_{6}=13,m13=41,m14=43{m}_{13}=41,{m}_{14}=43,m7=17,m8=19{m}_{7}=17,{m}_{8}=19m15=47,m16=53{m}_{15}=47,{m}_{16}=534(10)4\left(10)m1=2,m2=3{m}_{1}=2,{m}_{2}=3,379659m11=31,m12=37{m}_{11}=31,{m}_{12}=37,549m3=5,m4=7{m}_{3}=5,{m}_{4}=7,m13=41,m14=43{m}_{13}=41,{m}_{14}=43,m5=11,m6=13{m}_{5}=11,{m}_{6}=13,m15=47,m16=53{m}_{15}=47,{m}_{16}=53,m7=17,m8=19{m}_{7}=17,{m}_{8}=19,m17=59,m18=61{m}_{17}=59,{m}_{18}=61,m9=23,m10=29{m}_{9}=23,{m}_{10}=29m19=67{m}_{19}=67The set {mj}OPT{\left\{{m}_{j}\right\}}_{{\rm{OPT}}}from kkcontrol bases of residue classes, determined from condition (3): (3)ΔV≥VRC(k).\Delta V\ge {V}_{{\rm{RC}}}^{(k)}.In conventional units, the number of CS equipment VPNS(n+k){V}_{{\rm{PNS}}}^{(n+k)}in the RNC is determined by the following equation: VPNS(n+k)=VPNS(n)+VPNS(k),{V}_{{\rm{PNS}}}^{(n+k)}={V}_{{\rm{PNS}}}^{(n)}+{V}_{{\rm{PNS}}}^{(k)},where VPNS(n)=∑i=1nαi{V}_{{\rm{PNS}}}^{(n)}={\sum }_{i=1}^{n}{\alpha }_{i}is a conditional relative number of CS equipment reduced to one binary digit of CS in the RNS without redundancy (the number of binary bits in the representation of all RNS information modules).VRNS(k)=∑i=n+1n+kαi{V}_{{\rm{RNS}}}^{(k)}={\sum }_{i=n+1}^{n+k}{\alpha }_{i}is an additional conditional relative quantity of the CS equipment, reduced to one bit of the digits of the redundant CS in the RNS, generated by introducing kkcontrol bases (the number of binary bits in the representation of all RNS control modules);αi=[log2(mi−1)]+1{\alpha }_{i}={[}{\log }_{2}({m}_{i}-1)]+1is a number of bits required to represent the base by modulo mi{m}_{i}RNS. The values n, k denote the number of information and control bases of the RNS, respectively. Set of kkcontrol bases of the RNS, is determined from the following condition: ΔV≥VRNS(k),\Delta V\ge {V}_{{\rm{RNS}}}^{(k)},where ΔV=V2PNS(l)−VRNS(n)(ΔV=V3PNS(l)−VRNS(n)),\Delta V={V}_{2{\rm{PNS}}}^{(l)}-{V}_{{\rm{RNS}}}^{(n)}(\Delta V={V}_{3{\rm{PNS}}}^{(l)}-{V}_{{\rm{RNS}}}^{(n)}),or from the following condition: VPNS(l)≥VRNS(n+k).{V}_{{\rm{PNS}}}^{(l)}\ge {V}_{{\rm{RNS}}}^{(n+k)}.The conditional relative amount of equipment for duplicated and tripled ll- byte (l=1…4)\left(l=1\ldots 4)CS in the PNS is determined by the expression V2PNS(l)=2⋅8⋅l{V}_{2{\rm{PNS}}}^{(l)}=2\cdot 8\cdot land V3PNS(l)=3⋅8⋅l{V}_{3{\rm{PNS}}}^{(l)}=3\cdot 8\cdot l, respectively.When conducting a comparative analysis of reliability for CS in the PNS, we will take duplicated and triple CS of the same type of CS. In this case, the probability of fault-tolerant operation for such CS will be determined, respectively, by the following well-known expressions: (4)P2PNS(l)(t)=1−[1−P0PNS(l)(t)]2,{P}_{2{\rm{PNS}}}^{(l)}\left(t)=1-{{[}1-{P}_{0{\rm{PNS}}}^{(l)}\left(t)]}^{2},(5)P3PNS(l)(t)=1−[1−P0PNS(l)(t)]3,{P}_{3{\rm{PNS}}}^{(l)}\left(t)=1-{{[}1-{P}_{0{\rm{PNS}}}^{(l)}\left(t)]}^{3},where P0PNS(l)(t)=e−8lλEt,{P}_{0{\rm{PNS}}}^{(l)}\left(t)={{\rm{e}}}^{-8l{\lambda }_{E}t},is a probability of fault-tolerant operation of redundant ll- byte positional CS; λE{\lambda }_{E}is a failure rate of a unit of the conditional equipment of the CS, assigned to one binary digit of the characters grid of the CS.The amount of equipment for a duplicate CS in the PNS is determined by the following expression: V2PNS(l)=2⋅8⋅l,{V}_{2{\rm{PNS}}}^{(l)}=2\cdot 8\cdot l,and for a triple CS is determined as V3PNS(l)=3⋅8⋅l.{V}_{3{\rm{PNS}}}^{(l)}=3\cdot 8\cdot l.Based on the data of Tables 1 and 2 and using expressions (6) (6)PRC(n/k)(t)=∑i=0kCn+kiPin+k−i(t)∑j=0i(−1)iCijPij(t),{P}_{{\rm{RC}}}^{\left(n\hspace{0.1em}\text{/}\hspace{0.1em}k)}\left(t)=\mathop{\sum }\limits_{i=0}^{k}{C}_{n+k}^{i}{{P}_{i}}^{n+k-i}\left(t)\mathop{\sum }\limits_{j=0}^{i}{(-1)}^{i}{C}_{i}^{j}{P}_{i}^{j}\left(t),we obtain expressions for determining the probability of fault-tolerance operation of the CS in the RNS for the values l=1…4l=1\ldots 4of byte digit (Table 3).Cn+ki=(n+k)!i!⋅(n+k−i)!{C}_{n+k}^{i}=\frac{\left(n+k)\hspace{0.1em}\text{&#x0021;}}{i\text{&#x0021;}\hspace{0.1em}\cdot \left(n+k-i)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}is a number of combinations of n+kn+kelements by ii.Table 3Mathematical expressions for calculating the values of the fault-tolerant operation probability of the CS in RNSIn relation to the duplicated CS in the PNSIn relation to the triple CS in the PNSPl=1RC(4/1)(t)=∑i=01C5iPi5−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=1}^{{\rm{RC}}}}^{\left(4\hspace{0.1em}\text{/}\hspace{0.1em}1)}\left(t)={\sum }_{i=0}^{1}{C}_{5}^{i}{{P}_{i}}^{5-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t)Pl=1RC(4/3)(t)=∑i=03C7iPi7−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=1}^{{\rm{RC}}}}^{\left(4\hspace{0.1em}\text{/}\hspace{0.1em}3)}\left(t)={\sum }_{i=0}^{3}{C}_{7}^{i}{{P}_{i}}^{7-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t)Pl=2RC(6/2)(t)=∑i=02C8iPi8−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=2}^{{\rm{RC}}}}^{\left(6\hspace{0.1em}\text{/}\hspace{0.1em}2)}\left(t)={\sum }_{i=0}^{2}{C}_{8}^{i}{{P}_{i}}^{8-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t)Pl=2RC(6/5)(t)=∑i=05C11iPi11−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=2}^{{\rm{RC}}}}^{\left(6\hspace{0.1em}\text{/}\hspace{0.1em}5)}\left(t)={\sum }_{i=0}^{5}{C}_{11}^{i}{{P}_{i}}^{11-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t)Pl=3RC(8/4)(t)=∑i=04C12iPi12−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=3}^{{\rm{RC}}}}^{\left(8\hspace{0.1em}\text{/}\hspace{0.1em}4)}\left(t)={\sum }_{i=0}^{4}{C}_{12}^{i}{{P}_{i}}^{12-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t)Pl=3RC(8/8)(t)=∑i=08C16iPi16−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=3}^{{\rm{RC}}}}^{\left(8\hspace{0.1em}\text{/}\hspace{0.1em}8)}\left(t)={\sum }_{i=0}^{8}{C}_{16}^{i}{{P}_{i}}^{16-i}\mathop{\sum }\limits_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t)Pl=4RC(10/4)(t)=∑i=04C14iPi14−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=4}^{{\rm{RC}}}}^{\left(10\hspace{0.1em}\text{/}\hspace{0.1em}4)}\left(t)={\sum }_{i=0}^{4}{C}_{14}^{i}{{P}_{i}}^{14-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t).Pl=4RC(10/9)(t)=∑i=09C19iPi19−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=4}^{{\rm{RC}}}}^{\left(10\hspace{0.1em}\text{/}\hspace{0.1em}9)}\left(t)={\sum }_{i=0}^{9}{C}_{19}^{i}{P}_{i}^{19-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t).Formulas similar to expression (6) are used to determine the probability of fault-tolerant operation of a CS in RNS in the case of a sliding redundancy mode with a loaded reserve with nninformation and kkcontrol computing paths. In this case, the information and control paths of the CS in RNS are considered equally reliable. For example, in a study of Polovko [24] there is a formula 7.5 (p. 319), as well as a similar formula in another study of Polovko [25].It should be noted that in accordance with the data of Tables 1 and 2, Tables A1 and A2 (Appendix) presents the dynamic ranges (the volume of the range of possible numbers) of the set of information D1{D}_{1}and control D2{D}_{2}RNS modules.Based on the data in Tables 1 and 2, the article synthesizes structures of fault-tolerance CS in RNS for l=1…4l=1\ldots 4byte characters grid (Figures 1, 2, 3, 4).Figure 1Block diagram of CS in RNS for value l=1l=1.Figure 2Block diagram of CS in RNS for value l=2l=2.Figure 3Block diagram of CS in RNS for value l=3l=3.Figure 4Block diagram of CS in RNS for value l=4l=4.Figure 1 presents the CS structure for a single-byte (l=1l=1) digit (8 binary digits). The structure of the CS contains four (n=4n=4) information computational paths and three (k=3k=3) control computational paths. In case of failure of the information computational paths, control computational paths are connected in its place. Tables 1 and 2 present sets of specific information and control modules of the RNS. Figure 2 presents the CS structure for a two-byte (l=2l=2) digit (16 binary digits). The structure of the CS contains 6 (n=6n=6) information computational paths and 5 (k=5k=5) control computational paths. In case of failure of the information computational paths, control computational paths are connected in its place. Tables 1 and 2 present sets of specific information and control modules of the RNS. Figure 3 presents the CS structure for a three-byte (l=3l=3) digit (24 binary bits). The structure of the CS contains eight (n=8n=8) information computational paths and eight (k=8k=8) control computational paths. In case of failure of the information computational paths, control computational paths are connected in its place. Tables 1 and 2 present sets of specific information and control modules of the RNS. Figure 4 presents the CS structure for a four-byte (l=4l=4) character grid (32 binary bits). The structure of the CS contains eight (n=10n=10) information computational paths and eight (k=9k=9) control computational paths. In case of failure of the information computational paths, control computational paths are connected in its place. Tables 1 and 2 present sets of specific information and control modules of the RNS.The structure of the CS in the RNS contains nninformation modules and kkcontrol modules. In Figures 1–4, a simplified structure of CSs in RNS for a one-byte (l=1l=1) character grid (8 binary digits), two-byte (l=2l=2) characters grid (16 binary digits), three-byte (l=3l=3) bit grid (24 binary bits) and a four-byte (l=4l=4) character grid (32 bits). Tables 1 and 2 present sets of specific nninformational (number of information computing paths) and kkcontrol modules of the RNS. In the article, the CS in the RNS is considered from the standpoint of the backup system in the PNS operating in the sliding redundancy mode with a loaded reserve. In this case, information and control computing paths play the role of the main and backup elements of the redundant system in the PNS respectively. The following basic assumptions are made for the functioning of the CS in the RNS. –Failures of computing paths satisfy the conditions of the simplest flow of failures. This is due to the fact that the simplest flow of CS failures was theoretically substantiated, confirmed experimentally and provided with information about the failure rate of the existing CS elements.–A switching device that performs the operations of determining failed computing paths and connecting in their place ideally working ones, i.e., the probability of failure-free operation of the switch is equal to one. On simplified structural diagrams (Figures 1–4), the CS in the RNS does not show the switching device.–Since the article deals with an ordered (mi<mi+1{m}_{i}\lt {m}_{i+1}) RNS, an arbitrary operable control computing path can be connected instead of any failed information computing path.–Recovery of the failed computing paths of the CS in the RNS is not provided.Figures 1–4 show the simplified structural diagrams of the CS in the RNS. In the general case, the CS in the RNS consists of n+kn+kcomputing paths, which operate independently of each other in parallel in time (RNS properties). For a specific case, we will make explanations using the example of Figure 1. Figure 1 presents CS in RNS implements calculations for a single-byte (l=1l=1or 8 binary digits) character grid. The CS operates with numbers in the numerical range from 0 to 28−1{2}^{8}-1or from 0 to 255. Information (n=4n=4) computing paths by modules m1=3{m}_{1}=3, m2=4{m}_{2}=4, m3=5{m}_{3}=5and m4=7{m}_{4}=7(Tables 1 or 2) ensure the operation of the CS in the RNS in the given 0 to 28−1{2}^{8}-1numerical range, as 28<m1m2m3m4=420{2}^{8}\lt {m}_{1}{m}_{2}{m}_{3}{m}_{4}=420. Figure 1 shows four (n=4n=4) information inputs (In. 1 – In. 4) of the CS to the information computational paths of which, through the corresponding RNS modules, the processed numbers are received. At the exits Out. 1-Out. 4 KS we obtain the result of the operation. Reserve computing paths 5–7 modulo m5=11{m}_{5}=11, m6=13{m}_{6}=13and m7=17{m}_{7}=17(Tables 1 or 2), respectively, provide an efficient state of the CS in the RNS in case of failure of any information computing path. In this case, the sequence of connecting backup computing paths 5–7 corresponds to the number of the control computing paths, as mi<mi+1{m}_{i}\lt {m}_{i+1}. The fulfillment of the condition mi<mi+1{m}_{i}\lt {m}_{i+1}makes it possible to ensure the possibility of the functioning of the CS in a given numerical range from 0 to 28−1{2}^{8}-1.In Figure 1, information and control computational paths are naturally included in the structure of the CS in the RNS. The location of the computing path of the CS in the RNS (Figure 1) only emphasizes the differences between information and control modifications.In accordance with this path mathematical expressions were first derived for calculating the values of the fault-tolerant operation probability of the CS in the RNSs (Table 3).Using the probability formulas for the fault-tolerant operation of the CS in the RNS with respect to the triple CS in the PNS (Table 3), as well as expression 6, the reliability of the CS in RNS was calculated. Table 4 presents the results of the calculations of the CS reliability in the PNS and CS reliability in the in RNS.Table 4The results of the calculations of the reliability of CS in the PNS and the RCPNSRCllλEt{\lambda }_{E}tOne CSTriplePermanent reservationDynamic reservationl=1l=110−31{0}^{-3}0.9920320.9999990.9999830.99999910−21{0}^{-2}0.9231160.9995460.9983170.99999510−11{0}^{-1}0.4493290.8330150.8563380.9157462×10−12\times 1{0}^{-1}0.2018970.4916330.5745970.521476l=2l=210−31{0}^{-3}0.9841270.9999960.9999671.00000010−21{0}^{-2}0.8521440.9967670.9967170.99999910−11{0}^{-1}0.2018970.4916320.7328310.9371412×10−12\times 1{0}^{-1}0.0407620.1173690.3225720.471169l=3l=310−31{0}^{-3}0.97612860.9999860.9999611.00000010−21{0}^{-2}0.7866280.9902850.9960290.99999910−11{0}^{-1}0.907170.2482110.6339950.9170942×10−12\times 1{0}^{-1}0.0082290.0244860.1766430.261531l=4l=410−31{0}^{-3}0.9685060.9999680.9999481.00000010−21{0}^{-2}0.7261490.9794620.9946181.00000010−11{0}^{-1}0.0407620.1173690.5295790.9244192×10−12\times 1{0}^{-1}0.0016610.0049760.0896110.2091705ConclusionBased on the research results presented in this article, the following main conclusions can be drawn. (1)Based on the application of the active fault-tolerant methods, the article presents the designed structures of the CS in the RNS for l=1…4l=1\ldots 4byte system (Figures 1–4). In accordance with which (on the basis of the initial data, Tables 1 and 2), mathematical expressions were first derived for calculating the values of the fault-tolerant operation probability of the CS in the RNS (Table 3). Calculation and comparative analysis of the reliability of the CS in the RNS and the widely used in the PNSs with triple computing structure, which confirmed the high efficiency of the use of the NCS (Table 4). Studies have shown that, first, the use of the active fault-tolerant method (dynamic reservation) in the in RNS provides higher reliability than that widely used in the PNS method of tripling of the same type of CS, as well as provide higher reliability than the passive fault-tolerant method (continuous redundancy) in the in RNS. Second, with an increase in the CS digits, which is characteristic of the modern tendency in the development of computing facilities, the efficiency of using NCS in RNS increases.(2)Theoretically, the increase in the reliability of the CS in RNS is explained as follows. First, based on the properties of independence and equality of the remainders of the number, the totality of which determines the NCS, the structure of the CS in RNS is a type of structural redundant system from the computing path according to the corresponding modules mi{m}_{i}of the residue classes. Each of the elements (computing paths) of this redundant system (CS) operates independently of each other and in parallel in time according to its own separate module. This in the PNS is equivalent to reserving the smaller nodes of the CS. Second, in accordance with the provisions of the general theory (from the point of view of increasing reliability), it is advisable to reserve small nodes and blocks of a complex system, since the failure rate of elements is always less than the failure rate of large nodes or the entire large system. The results of a comparative reliability analysis showed that the probability of a fault-tolerant operation of the CS in RNS is higher than the probability of the fault-tolerant operation of the tripled majority structure in the PNS.(3)There is an assumption that the data encoding in RNS solves the problem of optimal backup, i.e., the creation of CS based on the use of residue classes provides it with maximum reliability under restrictions on its overall dimensions. However, this assumption requires a separate strict proof. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Computer Science de Gruyter

Designing of fault-tolerant computer system structures using residue number systems

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© 2022 Victor Krasnobayev et al., published by De Gruyter
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Abstract

1IntroductionCurrently, the main directions of computer systems and components (CSC) development in a positional number system (PNS) are improving the speed of implementation of arithmetic operations (including integer arithmetic operations) and reliability of their functioning [1,2,3].The complexity, scale and volume of various computational tasks and tasks of managing large and complex information systems which are solved by computer system (CS) necessitate the expansion of functions and capabilities of the CS. It leads to an increase in the number and complexity of computing equipment and system equipment, and in the complexity of their software [4,5]. This, in turn, necessitates the adoption of additional measures to ensure the high reliability of the CS and, first of all, the need to take additional measures to ensure the high fault tolerance of the CSC [6,7].2State-of-the-artThere are two main principles of increasing the reliability of the CS that operates in the PNS [8,9,10]: increasing reliability of individual elements of the CS (using a new base element) and introducing various types of redundancies. Since the reliability of logical elements is mainly determined by the current level of technology development, it is obvious that the introduction of redundancy when using any available element base is the most effective way to increase the reliability of the computer-controlled system [11,12].The listed measures for introducing redundancy ensure the availability of fault-tolerant properties in CS. The presence of fault-tolerance properties in the CS allows us to increase the reliability of its functioning. The fault-tolerant property [7] provides the ability to perform specified computational functions after failures, both by reducing, within acceptable limits, any indicators of the quality of functioning (e.g., by gradual degradation), and without deteriorating the quality of functioning of the computer-controlled system [2,13,14].The following is shown in the literature on this subject [2,15,16]. First, the use of binary PNS as the number system of the CS does not allow us to dramatically increase productivity and reliability. Second, there are results of basic research, which show possibility of a significant increase in the speed of implementation of integer arithmetic operations: addition, multiplication and subtraction by using a non-positional residue number system (RNS). This is achieved through the use of the properties of numbering systems, namely, independence, equal rights and low rate of transfer of residuals, which determine the non-positional code structure (NCS). The first and second properties of RNSs allow us to parallelize arithmetic calculations at the level of decomposition of residue numbers. Also, they provide an opportunity to realize spatial diversity of data elements with the possibility of their subsequent asynchronous independent processing. The use of the third properties of residue classes allow us to implement tabular methods for performing integer arithmetic operations of the basic set and implement polynomial functions with a single-cycle sample of the result of a modular operation. Thus, the use of RNS as number system of CS significantly (compared with PNS) increases the speed of the CSC.Suppose that at the design stage it is necessary to provide the needed (predetermined) level of fault-tolerance of a CS. It is possible to increase (ensure) the reliability if the system has a certain property, the use of which will allow it to be done [6,17]. Such property is defined and called fault-tolerance [7,18,19]. The presence of the fault tolerance property of the CS can be ensured both through the application of the method of passive fault tolerance (using the method of constant structural redundancy) [6,20] and through the method of active fault tolerance [21] (dynamic backup method) by introducing, in addition to the main computing paths CS in RNS, backup computing paths.In refs [14,22], it was noted that the use of passive fault-tolerant methods provides increased reliability of the CS in RNS in comparison with the majority turned computing system widely used in the PNSs. However, the lack of research results on the possibility of using residue classes to increase the reliability of the CS based on the application of active fault-tolerant methods limits the possibility of a comprehensive consideration of the task of ensuring the reliability of the functioning of the CSC. The purpose of the article is to design structures of a fault-tolerant CS in RNS in the case of applying the active fault-tolerant method.3Statement of the optimal redundancy problemBased on the properties of the RNS, the independence, equality and low number of digits of the residues representing the number of the structure of the CS in RNS are based on the use of the idea of structural redundancy in the PNS. In this case, the computing paths, according to the corresponding modules of the RNS, play the role of the main elements of the redundant system, and the control computing paths of the RNS play the role of the backup elements of the redundant system. Theoretically, the increase in the reliability of the CS in the RNS is explained as follows. First, based on the properties of independence and equality of the remainders, the totality of which determines the NCS, the structure of the CS in the RNS is a kind of a structural redundant system, consisting of computing paths according to the corresponding RNS modules. Each of the elements (computing paths) of this redundant system (CS) operates independently of each other and in parallel in time and each in its own separate module. This in the PNS is equivalent to the fact that smaller nodes of the CS are reserved. Second, in accordance with the provisions of the general reliability theory (from the point of view of increasing reliability), it is advisable to reserve small nodes and blocks of a complex system, since the failure rate of elements is always less than the failure rate of large nodes or the entire large system as a whole. The probability of fault-tolerant operation of the CS in the RNS is higher than the probability of fault-tolerant operation of the methods of increasing the reliability of duplicated and tripled majority computer structures widely used in practice. For the active fault-tolerant methods, the task of designing a fault-tolerant CS operating in the RNS is verbally formulated as follows: it is necessary to determine the optimal set {mj}OPT{\left\{{m}_{j}\right\}}_{{\rm{OPT}}}of modules from possible control mn+1,mn+2,…,mn+k{m}_{n+1},{m}_{n+2},\ldots ,{m}_{n+k}, at which reliability (fault tolerance) PRC(n/k)(t){P}_{{\rm{RC}}}^{(n\hspace{0.1em}\text{/}\hspace{0.1em}k)}\left(t)CS in the RNS will be maximum. However, there are limits on the amount of equipment VRC(n+k){V}_{{\rm{RC}}}^{\left(n+k)}CS in the RNS.To get the optimal set {mj}OPT{\left\{{m}_{j}\right\}}_{{\rm{OPT}}}of bases, it is necessary to formulate and solve the inverse problem of optimal reservation in RNS. If necessary, for the application of the active fault-tolerant method (application of the dynamic structural redundancy method), the inverse optimal reservation problem in RNS is mathematically formulated as follows (equation (1)): (1)PRC(n+z)(t)[t=const]→max;(VPNS(l)≥VRC(n+k)),\left\{\begin{array}{c}{P}_{{\rm{RC}}}^{\left(n+z)}\left(t){[}t={\rm{const}}]\to \max ;\\ ({V}_{{\rm{PNS}}}^{(l)}\ge {V}_{{\rm{RC}}}^{(n+k)}),\end{array}\right.where kkis a maximum possible number of control bases mn+1,mn+2,…,mn+k{m}_{n+1},{m}_{n+2},\ldots ,{m}_{n+k}(the number of CS control computing paths in residue classes); {m(n+z)}OPT(z=1…k){\{{m}_{\left(n+z)}\}}_{{\rm{OPT}}}\hspace{0.33em}\left(z=1\ldots k)is an optimal (providing a minimum amount of equipment VRC(n+k){V}_{{\rm{RC}}}^{(n+k)}CS in RNS) set of control bases (for a given value l of the CS character grid), the simultaneous use of which ensures the maximum fault-tolerance value PRC(n/z)(t){P}_{{\rm{RC}}}^{(n\hspace{0.1em}\text{/}\hspace{0.1em}z)}\left(t)CS in residue classes (RC).In this case, the following condition must be met: (2)VRC(k)≤VPNS(l)−VRC(n).{V}_{{\rm{RC}}}^{(k)}\le {V}_{{\rm{PNS}}}^{(l)}-{V}_{{\rm{RC}}}^{(n)}.It is convenient to solve the inverse optimal reservation problem in RNS with restriction by the method of coordinate wise steepest descent or by the method of dynamic programming [23].4Structures of CSs functioning in RNSLet us consider examples of constructing CS structures in RNS for l=1…4l=1\ldots 4. Tables 1 and 2 show the sets of information nnand control kkbases of the residue classes for the values of llof the corresponding characters grids l=1…4l=1\ldots 4of the CS. The data on the conditional relative amount of equipment of the computer station in RNS are presented, as well as the data on the conditional relative quantity of the equipment of CS in the PNS, reduced to one binary digit of the length llof the grid, where VRC(n)=∑i=1nαi,{V}_{{\rm{RC}}}^{(n)}=\mathop{\sum }\limits_{i=1}^{n}{\alpha }_{i},is a conditional relative amount of the equipment of the CS, reduced to one binary digit without excess CS in the RNS.VRC(k)=∑i=n+1n+kαi,{V}_{{\rm{RC}}}^{(k)}=\mathop{\sum }\limits_{i=n+1}^{n+k}{\alpha }_{i},is an additional conditional relative amount of CS equipment, reduced to one binary digit of CS in RNS, formed by the introduction of control bases. αi=[log2(mi−1)]+1{\alpha }_{i}={[}{\log }_{2}({m}_{i}-1)]+1is a number of bits needed to represent the mi{m}_{i}of the residue classes. ⌊X⌋\lfloor X\rfloor is an integer part of the number XXthat does not exceed it (rounding down). For example, [7.9] = 7. V0PNS(l){V}_{0{\rm{PNS}}}^{(l)}is a conditional relative amount of equipment without CS excess in the PNS. V2PNS(l){V}_{2{\rm{PNS}}}^{(l)}is a conditional relative amount of equipment of duplicated CS in the PNS. V3PNS(l){V}_{3{\rm{PNS}}}^{(l)}is a conditional relative amount of equipment of triple CS in the PNS.Table 1The sets of information nnand control kkbases of the residue classes, 2PNSl(n)l\left(n){mi},i=1…n\{{m}_{i}\},\hspace{0.33em}i=1\ldots nVRC(n){V}_{{\rm{RC}}}^{(n)}V2PNS(l){V}_{2{\rm{PNS}}}^{(l)}ΔV\Delta V{mj},j=n+1…n+k\{{m}_{j}\},\hspace{0.33em}j=n+1\ldots n+kVRC(k){V}_{{\rm{RC}}}^{(k)}kk1(4)m1=3,m2=4{m}_{1}=3,{m}_{2}=4,10166m5=11{m}_{5}=1141m3=5,m4=7{m}_{3}=5,{m}_{4}=72(6)2\left(6)m1=2,m2=5{m}_{1}=2,{m}_{2}=5,193213m7=17,m8=19{m}_{7}=17,{m}_{8}=19102m3=7,m4=9{m}_{3}=7,{m}_{4}=9,m5=11,m6=13{m}_{5}=11,{m}_{6}=133(8)m1=3,m2=4{m}_{1}=3,{m}_{2}=4,284820m9=23,m10=29,{m}_{9}=23,{m}_{10}=29,214m3=3,m4=7{m}_{3}=3,{m}_{4}=7,m11=31,m12=37{m}_{11}=31,{m}_{12}=37m5=11,m6=13{m}_{5}=11,{m}_{6}=13,m7=17,m8=19{m}_{7}=17,{m}_{8}=194(10)m1=2,m2=3{m}_{1}=2,{m}_{2}=3,376427m11=31,m12=37{m}_{11}=31,{m}_{12}=37,234m3=5,m4=7{m}_{3}=5,{m}_{4}=7,m13=41,m14=43{m}_{13}=41,{m}_{14}=43m5=11,m6=13{m}_{5}=11,{m}_{6}=13,m7=17,m8=19{m}_{7}=17,{m}_{8}=19,m9=23,m10=29{m}_{9}=23,{m}_{10}=29Table 2The sets of information nnand control kkbases of the residue classes, 3PNSl(n)l\left(n){mi},i=1…n\{{m}_{i}\},\hspace{0.33em}i=1\ldots nVRC(n){V}_{{\rm{RC}}}^{(n)}V2PNS(l){V}_{2{\rm{PNS}}}^{(l)}ΔV\Delta V{mj},j=n+1…n+k\{{m}_{j}\},\hspace{0.33em}j=n+1\ldots n+kVRC(k){V}_{{\rm{RC}}}^{(k)}kk1(4)m1=3,m2=4,{m}_{1}=3,{m}_{2}=4,102414m5=11,m6=13{m}_{5}=11,{m}_{6}=13,133m3=5,m4=7{m}_{3}=5,{m}_{4}=7m7=17{m}_{7}=172(6)m1=2,m2=5{m}_{1}=2,{m}_{2}=5,194829m7=17,m8=19{m}_{7}=17,{m}_{8}=19,255m3=7,m4=9{m}_{3}=7,{m}_{4}=9,m9=23,m10=29{m}_{9}=23,{m}_{10}=29,m5=11,m6=13{m}_{5}=11,{m}_{6}=13m11=31{m}_{11}=313(8)m1=3,m2=4{m}_{1}=3,{m}_{2}=4,287244m9=23,m10=29{m}_{9}=23,{m}_{10}=29,458m3=3,m4=7{m}_{3}=3,{m}_{4}=7,m11=31,m12=37{m}_{11}=31,{m}_{12}=37,m5=11,m6=13{m}_{5}=11,{m}_{6}=13,m13=41,m14=43{m}_{13}=41,{m}_{14}=43,m7=17,m8=19{m}_{7}=17,{m}_{8}=19m15=47,m16=53{m}_{15}=47,{m}_{16}=534(10)4\left(10)m1=2,m2=3{m}_{1}=2,{m}_{2}=3,379659m11=31,m12=37{m}_{11}=31,{m}_{12}=37,549m3=5,m4=7{m}_{3}=5,{m}_{4}=7,m13=41,m14=43{m}_{13}=41,{m}_{14}=43,m5=11,m6=13{m}_{5}=11,{m}_{6}=13,m15=47,m16=53{m}_{15}=47,{m}_{16}=53,m7=17,m8=19{m}_{7}=17,{m}_{8}=19,m17=59,m18=61{m}_{17}=59,{m}_{18}=61,m9=23,m10=29{m}_{9}=23,{m}_{10}=29m19=67{m}_{19}=67The set {mj}OPT{\left\{{m}_{j}\right\}}_{{\rm{OPT}}}from kkcontrol bases of residue classes, determined from condition (3): (3)ΔV≥VRC(k).\Delta V\ge {V}_{{\rm{RC}}}^{(k)}.In conventional units, the number of CS equipment VPNS(n+k){V}_{{\rm{PNS}}}^{(n+k)}in the RNC is determined by the following equation: VPNS(n+k)=VPNS(n)+VPNS(k),{V}_{{\rm{PNS}}}^{(n+k)}={V}_{{\rm{PNS}}}^{(n)}+{V}_{{\rm{PNS}}}^{(k)},where VPNS(n)=∑i=1nαi{V}_{{\rm{PNS}}}^{(n)}={\sum }_{i=1}^{n}{\alpha }_{i}is a conditional relative number of CS equipment reduced to one binary digit of CS in the RNS without redundancy (the number of binary bits in the representation of all RNS information modules).VRNS(k)=∑i=n+1n+kαi{V}_{{\rm{RNS}}}^{(k)}={\sum }_{i=n+1}^{n+k}{\alpha }_{i}is an additional conditional relative quantity of the CS equipment, reduced to one bit of the digits of the redundant CS in the RNS, generated by introducing kkcontrol bases (the number of binary bits in the representation of all RNS control modules);αi=[log2(mi−1)]+1{\alpha }_{i}={[}{\log }_{2}({m}_{i}-1)]+1is a number of bits required to represent the base by modulo mi{m}_{i}RNS. The values n, k denote the number of information and control bases of the RNS, respectively. Set of kkcontrol bases of the RNS, is determined from the following condition: ΔV≥VRNS(k),\Delta V\ge {V}_{{\rm{RNS}}}^{(k)},where ΔV=V2PNS(l)−VRNS(n)(ΔV=V3PNS(l)−VRNS(n)),\Delta V={V}_{2{\rm{PNS}}}^{(l)}-{V}_{{\rm{RNS}}}^{(n)}(\Delta V={V}_{3{\rm{PNS}}}^{(l)}-{V}_{{\rm{RNS}}}^{(n)}),or from the following condition: VPNS(l)≥VRNS(n+k).{V}_{{\rm{PNS}}}^{(l)}\ge {V}_{{\rm{RNS}}}^{(n+k)}.The conditional relative amount of equipment for duplicated and tripled ll- byte (l=1…4)\left(l=1\ldots 4)CS in the PNS is determined by the expression V2PNS(l)=2⋅8⋅l{V}_{2{\rm{PNS}}}^{(l)}=2\cdot 8\cdot land V3PNS(l)=3⋅8⋅l{V}_{3{\rm{PNS}}}^{(l)}=3\cdot 8\cdot l, respectively.When conducting a comparative analysis of reliability for CS in the PNS, we will take duplicated and triple CS of the same type of CS. In this case, the probability of fault-tolerant operation for such CS will be determined, respectively, by the following well-known expressions: (4)P2PNS(l)(t)=1−[1−P0PNS(l)(t)]2,{P}_{2{\rm{PNS}}}^{(l)}\left(t)=1-{{[}1-{P}_{0{\rm{PNS}}}^{(l)}\left(t)]}^{2},(5)P3PNS(l)(t)=1−[1−P0PNS(l)(t)]3,{P}_{3{\rm{PNS}}}^{(l)}\left(t)=1-{{[}1-{P}_{0{\rm{PNS}}}^{(l)}\left(t)]}^{3},where P0PNS(l)(t)=e−8lλEt,{P}_{0{\rm{PNS}}}^{(l)}\left(t)={{\rm{e}}}^{-8l{\lambda }_{E}t},is a probability of fault-tolerant operation of redundant ll- byte positional CS; λE{\lambda }_{E}is a failure rate of a unit of the conditional equipment of the CS, assigned to one binary digit of the characters grid of the CS.The amount of equipment for a duplicate CS in the PNS is determined by the following expression: V2PNS(l)=2⋅8⋅l,{V}_{2{\rm{PNS}}}^{(l)}=2\cdot 8\cdot l,and for a triple CS is determined as V3PNS(l)=3⋅8⋅l.{V}_{3{\rm{PNS}}}^{(l)}=3\cdot 8\cdot l.Based on the data of Tables 1 and 2 and using expressions (6) (6)PRC(n/k)(t)=∑i=0kCn+kiPin+k−i(t)∑j=0i(−1)iCijPij(t),{P}_{{\rm{RC}}}^{\left(n\hspace{0.1em}\text{/}\hspace{0.1em}k)}\left(t)=\mathop{\sum }\limits_{i=0}^{k}{C}_{n+k}^{i}{{P}_{i}}^{n+k-i}\left(t)\mathop{\sum }\limits_{j=0}^{i}{(-1)}^{i}{C}_{i}^{j}{P}_{i}^{j}\left(t),we obtain expressions for determining the probability of fault-tolerance operation of the CS in the RNS for the values l=1…4l=1\ldots 4of byte digit (Table 3).Cn+ki=(n+k)!i!⋅(n+k−i)!{C}_{n+k}^{i}=\frac{\left(n+k)\hspace{0.1em}\text{&#x0021;}}{i\text{&#x0021;}\hspace{0.1em}\cdot \left(n+k-i)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}is a number of combinations of n+kn+kelements by ii.Table 3Mathematical expressions for calculating the values of the fault-tolerant operation probability of the CS in RNSIn relation to the duplicated CS in the PNSIn relation to the triple CS in the PNSPl=1RC(4/1)(t)=∑i=01C5iPi5−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=1}^{{\rm{RC}}}}^{\left(4\hspace{0.1em}\text{/}\hspace{0.1em}1)}\left(t)={\sum }_{i=0}^{1}{C}_{5}^{i}{{P}_{i}}^{5-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t)Pl=1RC(4/3)(t)=∑i=03C7iPi7−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=1}^{{\rm{RC}}}}^{\left(4\hspace{0.1em}\text{/}\hspace{0.1em}3)}\left(t)={\sum }_{i=0}^{3}{C}_{7}^{i}{{P}_{i}}^{7-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t)Pl=2RC(6/2)(t)=∑i=02C8iPi8−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=2}^{{\rm{RC}}}}^{\left(6\hspace{0.1em}\text{/}\hspace{0.1em}2)}\left(t)={\sum }_{i=0}^{2}{C}_{8}^{i}{{P}_{i}}^{8-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t)Pl=2RC(6/5)(t)=∑i=05C11iPi11−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=2}^{{\rm{RC}}}}^{\left(6\hspace{0.1em}\text{/}\hspace{0.1em}5)}\left(t)={\sum }_{i=0}^{5}{C}_{11}^{i}{{P}_{i}}^{11-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t)Pl=3RC(8/4)(t)=∑i=04C12iPi12−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=3}^{{\rm{RC}}}}^{\left(8\hspace{0.1em}\text{/}\hspace{0.1em}4)}\left(t)={\sum }_{i=0}^{4}{C}_{12}^{i}{{P}_{i}}^{12-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t)Pl=3RC(8/8)(t)=∑i=08C16iPi16−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=3}^{{\rm{RC}}}}^{\left(8\hspace{0.1em}\text{/}\hspace{0.1em}8)}\left(t)={\sum }_{i=0}^{8}{C}_{16}^{i}{{P}_{i}}^{16-i}\mathop{\sum }\limits_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t)Pl=4RC(10/4)(t)=∑i=04C14iPi14−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=4}^{{\rm{RC}}}}^{\left(10\hspace{0.1em}\text{/}\hspace{0.1em}4)}\left(t)={\sum }_{i=0}^{4}{C}_{14}^{i}{{P}_{i}}^{14-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t).Pl=4RC(10/9)(t)=∑i=09C19iPi19−i∑j=0i(−1)jCijPij(t){P}_{{}_{l=4}^{{\rm{RC}}}}^{\left(10\hspace{0.1em}\text{/}\hspace{0.1em}9)}\left(t)={\sum }_{i=0}^{9}{C}_{19}^{i}{P}_{i}^{19-i}{\sum }_{j=0}^{i}{(-1)}^{j}{C}_{i}^{j}{P}_{i}^{j}\left(t).Formulas similar to expression (6) are used to determine the probability of fault-tolerant operation of a CS in RNS in the case of a sliding redundancy mode with a loaded reserve with nninformation and kkcontrol computing paths. In this case, the information and control paths of the CS in RNS are considered equally reliable. For example, in a study of Polovko [24] there is a formula 7.5 (p. 319), as well as a similar formula in another study of Polovko [25].It should be noted that in accordance with the data of Tables 1 and 2, Tables A1 and A2 (Appendix) presents the dynamic ranges (the volume of the range of possible numbers) of the set of information D1{D}_{1}and control D2{D}_{2}RNS modules.Based on the data in Tables 1 and 2, the article synthesizes structures of fault-tolerance CS in RNS for l=1…4l=1\ldots 4byte characters grid (Figures 1, 2, 3, 4).Figure 1Block diagram of CS in RNS for value l=1l=1.Figure 2Block diagram of CS in RNS for value l=2l=2.Figure 3Block diagram of CS in RNS for value l=3l=3.Figure 4Block diagram of CS in RNS for value l=4l=4.Figure 1 presents the CS structure for a single-byte (l=1l=1) digit (8 binary digits). The structure of the CS contains four (n=4n=4) information computational paths and three (k=3k=3) control computational paths. In case of failure of the information computational paths, control computational paths are connected in its place. Tables 1 and 2 present sets of specific information and control modules of the RNS. Figure 2 presents the CS structure for a two-byte (l=2l=2) digit (16 binary digits). The structure of the CS contains 6 (n=6n=6) information computational paths and 5 (k=5k=5) control computational paths. In case of failure of the information computational paths, control computational paths are connected in its place. Tables 1 and 2 present sets of specific information and control modules of the RNS. Figure 3 presents the CS structure for a three-byte (l=3l=3) digit (24 binary bits). The structure of the CS contains eight (n=8n=8) information computational paths and eight (k=8k=8) control computational paths. In case of failure of the information computational paths, control computational paths are connected in its place. Tables 1 and 2 present sets of specific information and control modules of the RNS. Figure 4 presents the CS structure for a four-byte (l=4l=4) character grid (32 binary bits). The structure of the CS contains eight (n=10n=10) information computational paths and eight (k=9k=9) control computational paths. In case of failure of the information computational paths, control computational paths are connected in its place. Tables 1 and 2 present sets of specific information and control modules of the RNS.The structure of the CS in the RNS contains nninformation modules and kkcontrol modules. In Figures 1–4, a simplified structure of CSs in RNS for a one-byte (l=1l=1) character grid (8 binary digits), two-byte (l=2l=2) characters grid (16 binary digits), three-byte (l=3l=3) bit grid (24 binary bits) and a four-byte (l=4l=4) character grid (32 bits). Tables 1 and 2 present sets of specific nninformational (number of information computing paths) and kkcontrol modules of the RNS. In the article, the CS in the RNS is considered from the standpoint of the backup system in the PNS operating in the sliding redundancy mode with a loaded reserve. In this case, information and control computing paths play the role of the main and backup elements of the redundant system in the PNS respectively. The following basic assumptions are made for the functioning of the CS in the RNS. –Failures of computing paths satisfy the conditions of the simplest flow of failures. This is due to the fact that the simplest flow of CS failures was theoretically substantiated, confirmed experimentally and provided with information about the failure rate of the existing CS elements.–A switching device that performs the operations of determining failed computing paths and connecting in their place ideally working ones, i.e., the probability of failure-free operation of the switch is equal to one. On simplified structural diagrams (Figures 1–4), the CS in the RNS does not show the switching device.–Since the article deals with an ordered (mi<mi+1{m}_{i}\lt {m}_{i+1}) RNS, an arbitrary operable control computing path can be connected instead of any failed information computing path.–Recovery of the failed computing paths of the CS in the RNS is not provided.Figures 1–4 show the simplified structural diagrams of the CS in the RNS. In the general case, the CS in the RNS consists of n+kn+kcomputing paths, which operate independently of each other in parallel in time (RNS properties). For a specific case, we will make explanations using the example of Figure 1. Figure 1 presents CS in RNS implements calculations for a single-byte (l=1l=1or 8 binary digits) character grid. The CS operates with numbers in the numerical range from 0 to 28−1{2}^{8}-1or from 0 to 255. Information (n=4n=4) computing paths by modules m1=3{m}_{1}=3, m2=4{m}_{2}=4, m3=5{m}_{3}=5and m4=7{m}_{4}=7(Tables 1 or 2) ensure the operation of the CS in the RNS in the given 0 to 28−1{2}^{8}-1numerical range, as 28<m1m2m3m4=420{2}^{8}\lt {m}_{1}{m}_{2}{m}_{3}{m}_{4}=420. Figure 1 shows four (n=4n=4) information inputs (In. 1 – In. 4) of the CS to the information computational paths of which, through the corresponding RNS modules, the processed numbers are received. At the exits Out. 1-Out. 4 KS we obtain the result of the operation. Reserve computing paths 5–7 modulo m5=11{m}_{5}=11, m6=13{m}_{6}=13and m7=17{m}_{7}=17(Tables 1 or 2), respectively, provide an efficient state of the CS in the RNS in case of failure of any information computing path. In this case, the sequence of connecting backup computing paths 5–7 corresponds to the number of the control computing paths, as mi<mi+1{m}_{i}\lt {m}_{i+1}. The fulfillment of the condition mi<mi+1{m}_{i}\lt {m}_{i+1}makes it possible to ensure the possibility of the functioning of the CS in a given numerical range from 0 to 28−1{2}^{8}-1.In Figure 1, information and control computational paths are naturally included in the structure of the CS in the RNS. The location of the computing path of the CS in the RNS (Figure 1) only emphasizes the differences between information and control modifications.In accordance with this path mathematical expressions were first derived for calculating the values of the fault-tolerant operation probability of the CS in the RNSs (Table 3).Using the probability formulas for the fault-tolerant operation of the CS in the RNS with respect to the triple CS in the PNS (Table 3), as well as expression 6, the reliability of the CS in RNS was calculated. Table 4 presents the results of the calculations of the CS reliability in the PNS and CS reliability in the in RNS.Table 4The results of the calculations of the reliability of CS in the PNS and the RCPNSRCllλEt{\lambda }_{E}tOne CSTriplePermanent reservationDynamic reservationl=1l=110−31{0}^{-3}0.9920320.9999990.9999830.99999910−21{0}^{-2}0.9231160.9995460.9983170.99999510−11{0}^{-1}0.4493290.8330150.8563380.9157462×10−12\times 1{0}^{-1}0.2018970.4916330.5745970.521476l=2l=210−31{0}^{-3}0.9841270.9999960.9999671.00000010−21{0}^{-2}0.8521440.9967670.9967170.99999910−11{0}^{-1}0.2018970.4916320.7328310.9371412×10−12\times 1{0}^{-1}0.0407620.1173690.3225720.471169l=3l=310−31{0}^{-3}0.97612860.9999860.9999611.00000010−21{0}^{-2}0.7866280.9902850.9960290.99999910−11{0}^{-1}0.907170.2482110.6339950.9170942×10−12\times 1{0}^{-1}0.0082290.0244860.1766430.261531l=4l=410−31{0}^{-3}0.9685060.9999680.9999481.00000010−21{0}^{-2}0.7261490.9794620.9946181.00000010−11{0}^{-1}0.0407620.1173690.5295790.9244192×10−12\times 1{0}^{-1}0.0016610.0049760.0896110.2091705ConclusionBased on the research results presented in this article, the following main conclusions can be drawn. (1)Based on the application of the active fault-tolerant methods, the article presents the designed structures of the CS in the RNS for l=1…4l=1\ldots 4byte system (Figures 1–4). In accordance with which (on the basis of the initial data, Tables 1 and 2), mathematical expressions were first derived for calculating the values of the fault-tolerant operation probability of the CS in the RNS (Table 3). Calculation and comparative analysis of the reliability of the CS in the RNS and the widely used in the PNSs with triple computing structure, which confirmed the high efficiency of the use of the NCS (Table 4). Studies have shown that, first, the use of the active fault-tolerant method (dynamic reservation) in the in RNS provides higher reliability than that widely used in the PNS method of tripling of the same type of CS, as well as provide higher reliability than the passive fault-tolerant method (continuous redundancy) in the in RNS. Second, with an increase in the CS digits, which is characteristic of the modern tendency in the development of computing facilities, the efficiency of using NCS in RNS increases.(2)Theoretically, the increase in the reliability of the CS in RNS is explained as follows. First, based on the properties of independence and equality of the remainders of the number, the totality of which determines the NCS, the structure of the CS in RNS is a type of structural redundant system from the computing path according to the corresponding modules mi{m}_{i}of the residue classes. Each of the elements (computing paths) of this redundant system (CS) operates independently of each other and in parallel in time according to its own separate module. This in the PNS is equivalent to reserving the smaller nodes of the CS. Second, in accordance with the provisions of the general theory (from the point of view of increasing reliability), it is advisable to reserve small nodes and blocks of a complex system, since the failure rate of elements is always less than the failure rate of large nodes or the entire large system. The results of a comparative reliability analysis showed that the probability of a fault-tolerant operation of the CS in RNS is higher than the probability of the fault-tolerant operation of the tripled majority structure in the PNS.(3)There is an assumption that the data encoding in RNS solves the problem of optimal backup, i.e., the creation of CS based on the use of residue classes provides it with maximum reliability under restrictions on its overall dimensions. However, this assumption requires a separate strict proof.

Journal

Open Computer Science – de Gruyter

Published: Jan 1, 2022

Keywords: fault-tolerant computer system; dynamic redundancy; optimal redundancy; residue number system

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