A Systematic Approach to Modeling and Analysis of Transient Faults in Logic Circuits

Transcription

1 A Systematic Approach to Modeing and Anaysis of Transient Fauts in Logic Circuits Natasa Miskov-Zivanov, Diana Marcuescu Department of Eectrica and Computer Engineering Carnegie Meon University Abstract With technoogy scaing, the occurrence rate of not ony singe, but aso mutipe transients resuting from a singe hit is increasing. In this work, we consider the effect of these mutipeevent transients on the outputs of ogic circuits. Our framework aows for the anaysis of soft errors in ogic circuits, incuding severa aspects: estimation of the effect of both singe and mutipe transient fauts on both combinationa and sequentia circuits, anaysis of the impact of mutipe fip-fop upsets in sequentia circuits, and anaysis of transient behavior of the soft error rate in the cyces foowing the hit. The proposed framework can be used to estimate the impact of transient fauts stemming not ony from radiation, but aso other physica phenomena. The resuts obtained using the proposed framework show that output error rates, resuting from mutipe-event transient or mutipe-bit upsets can vary across different circuits by severa orders of magnitude. 1. Introduction The scaing of device feature sizes, operating votages and design margins raises a great concern about the susceptibiity of circuits to transient fauts, which can be caused by different physica phenomena, such as high-energy partice hits originating from cosmic rays, capacitive couping, eectromagnetic interference, or power transients [10]. Transient fauts induced by radiation, aso caed Singe- Event Transients (SETs, are caimed to be a maor chaenge for future scaing [1] and have thus been examined by many researchers in recent years. An error that resuts from an SET (gitch or puse is most often referred to as soft error or a singe-event upset (SEU. The effect of soft errors is measured by the soft error rate (SER in FITs (faiure-in-time, which is defined as one faiure in 10 9 hours. In the past, soft errors used to be a concern ony in memories, thus resuting in widey used Error Correcting Codes (ECC mitigation techniques. However, these techniques may not onger be very efficient with the current and future technoogy nodes, due to the more often occurrence of a new faiure type, namey Mutipe-Bit Upset (MBU. The MBU is defined as severa adacent bit fais, simutaneousy induced by a unique partice hit. Up to five such tied up bit fais have been observed in a 130nm SRAM [5]. On the other hand, with the reduction of device dimensions and operating votage, the impact of radiation in ogic circuits is increasing and fast reaching the soft error rates in memories [11]. Therefore, SETs in ogic circuits are becoming an important reiabiity concern for future technoogy nodes. Furthermore, since the distances between unctions are decreasing with scaing, and the critica charge is reducing, the energy of radiation partices that is required to cause a mutipe transient faut is decreasing. The probabiity that a singe high energetic partice affects the output of more than one circuit node (Figure 1 is no onger negigibe [8] and if the two (or more affected nodes beong to different ogic gates, mutipe transient fauts (often referred to as Mutipe- Event Transients or METs can be generated and propagated to ogic circuit outputs. Hence, the evauation of accurate SER stemming from not ony singe transients, but aso mutipe transients becomes mandatory for very deep submicron technoogies. Therefore, as described above, the importance of reaistic and accurate proection of the SET- and MET-induced SER in ogic (combinationa and sequentia circuits is crucia to identifying the features needed for future reiabe high-performance microprocessors. In this work, we present an efficient and accurate methodoogy for the evauation of the impact of both singe and mutipe transient fauts in combinationa and sequentia circuits. The framework described here aows for unified treatment and probabiistic anaysis of important aspects of transient faut propagation in ogic circuits. It further provides the anaysis of the effects of transient fauts in the cyces foowing the hit as we as the impact of mutipe fip-fop upsets. The rest of this work is organized as foows. In Section 2, we describe previous work on soft error modeing and anaysis and briefy outine the contributions of our work. Section 3 provides an overview of the preiminaries of an SET anaysis in ogic circuits. The proposed approach to modeing and anaysis of mutipe transients in ogic circuits is described in Section 4. In Section 5, we describe the mode for mutipe fip-fop upsets that is incorporated into our framework, whie in Section 6 we present the mode for overa error computation. Finay, in Section 7 we show the experimenta resuts obtained using the proposed framework and with Section 8 we concude our work. 2. Reated work Among a transient fauts, radiation-induced fauts have received most of the attention in recent years, since they are considered as one of the maor barriers for future technoogy scaing [1]. Intensive research has been done so far in the area of modeing, anaysis, and protection for radiation-induced transient fauts [2],[3],[6],[7],[10],[11]. Since our focus is on modeing of these fauts and anayzing their effect on ogic circuits, we give a brief overview of the work reated to those aspects of transient fauts in the seque Singe-event transient anaysis One obvious approach to anayze the impact of transient fauts is to inect the faut into a given node of the circuit and simuate the circuit for different input vectors in order to find 1

2 whether the faut propagates. However, this approach becomes intractabe for arge circuits and arge number of inputs, and thus gives way to approximate approaches that use anaytica and symboic methods to evauate circuit susceptibiity to transient fauts. A number of methods have been proposed recenty to evauate the susceptibiity of combinationa ogic circuits to soft errors, among them severa symboic modes [6]. An exampe of such symboic modeing approach is the one that uses Binary Decision Diagrams (BDDs and Agebraic Decision Diagrams (ADDs to mode the propagation of transient fauts in ogic circuits [6]. This mode has been shown to be both efficient and accurate, and thus we incorporate its main ideas into our work. In contrast to existing methods for modeing soft error susceptibiity of combinationa circuits, sequentia circuits have received ess attention. A transient faut resuting from singe partice hit can affect outputs of a sequentia circuit during severa cock cyces. To consider this effect, the anaysis of the propagation of an SET through sequentia circuit is necessary for more than one cock cyce. An approach that tackes this issue in an efficient and accurate manner is described in [7] Mutipe-event transient anaysis The probem of METs has been addressed in the past, but it focused mosty on their effect in memories [4],[5]. Mutipe transients in ogic received very itte attention, unti recenty [8],[9]. Simiar to SETs, previous work that focused on METs in ogic circuits used, to the best of our knowedge, ony simuation. For exampe, the authors in [9] used simuation to estimate the sizes of mutipe transients resuting from a singe nucear reaction, as we as the impact of different gate input combinations on the output transient current. In case of approaches focusing on the effect of mutipe transients on the error rate of the overa circuit, previous work used simuations at either device [8] or circuit eve [4],[9] and the usua approach was to inect fauts at different nodes in the circuit, and then estimate the impact of those fauts on circuit outputs for different input combinations Paper contribution With respect to singe- and mutipe-event transients, the main contributions of this work, when compared to previous work, improve state-of-the-art by aowing for: Accurate and efficient modeing and anaysis of the impact of both SETs and METs in ogic (combinationa and sequentia circuits; Evauation of changes in error rates due to SETs and METs foowing the cyce when the transient faut occurred within the circuit; Evauation of the impact of mutipe fip-fop upsets in sequentia circuits. Figure 1. Schematic of a device with obique incident ange of partice hit causing doube transient (r = radius of the partice track. It is aso important to note that the specific parameters reated to radiation-induced transients (e.g., partice hit rate, ratio of effective hits are not directy part of the proposed framework, but instead are incuded as inputs to the framework. Thus, our framework can be appied to any type of a transient faut, irrespective of its origin. 3. Transient faut mode In this section, we briefy describe the main principes of generation and propagation of singe- and mutipe-event transients Faut generation When a high-energy charged partice passes through a semiconductor materia, it frees eectron-hoe pairs aong its path as it oses energy. Charge coection generay occurs within a few microns of the unction. The coected charge for the radiation-induced events in siicon can range from one to severa hundreds of fcs [3]. The device sensitivity to this excess charge is defined primariy by the node capacitance, operating votages, the strength of feedback or fanout transistors, a defining the amount of critica charge required to trigger a change in the data state. Critica charge for technoogy nodes beow 90nm decreases to 10fC [11]. When an energetic partice hits a device at an obique ange, there is a sma, but non-zero probabiity of disturbing more than one sensitive unction, as shown in Figure 1. The arger the partice track and the coser the unctions are, the arger the probabiity is for upsetting more than one unction [4]. The most probabe ocation of the occurrence of mutipe transients is at the outputs of neighboring gates. In other words, a gate and its fanin or fanout neighbors are possibe candidates for MET generation. Another possibiity woud be gates that have a common fanin or fanout neighbor. There are severa factors that need to be considered when modeing mutipe-event transients. First, the exact reationship between, for exampe, two METs generated by an energetic partice hitting two unctions, and the same partice affecting ony one unction cannot be determined in a straightforward manner. For exampe, given the partice hit, it is necessary to know how the charge coected by a singe unction, Q co, compares to the charge induced by the same energetic partice, but coected by two or more unctions (Q co,1, Q co,2,,q co,n. One possibe assumption is that: n Q co (1 Q co,i i=1 where the inequaity stems from the fact that the charge spread across severa nodes may resut in ess overa charge being coected. However, the exact reationship between the charge coected by a singe node, Q co and the sum of the charges coected by mutipe nodes is aso affected by the incident partice ange and the coection capacity of nodes. Next, the mutipe transients that can resut from a singe hit are not necessariy uniform and can be of different sizes. Even if we assume equaity in equation (3, the reationship between resuting gitch sizes is not the same as reationship between the coected charges, i.e.: D SET D MET,i n i=1 (2 2

3 a b c d Figure 2. HSPICE simuation resuts for a NAND gate with different oad and different coected charge combinations: a FO1, first gitch 10fC b FO1, first gitch 20fC, c FO2, first gitch 40fC, d first gate FO1, first gitch 20fC and second gate FO2. where D SET is the duration of the gitch resuting from Q co and D MET,i is the duration of the gitch in MET resuting from Q co,i. One possibe option is to use HSPICE simuations to determine this reationship in order to be abe to compare the impact of an SET to the impact of corresponding METs on the overa circuit reiabiity. We conducted HSPICE simuations of different gates, in order to determine the sizes of gitches resuting from a given coected charge. We show the resuts for a NAND gate in Figure 2, assuming different gate oad vaues (fan-out-of-1, FO1, and fan-out-of-2, FO2 and different coected charge (from 10 to 200fC. The origina curve in Figure 2 represents the duration of the gitch resuting from a given coected charge, whie the sum curve represents the sum of gitch durations, assuming one gitch resuts from a fixed charge: (a 10fC, (b 20fC, (c 40fC, and (d 20fC. The coected charge for the second gitch is varied, starting with the same vaue as for the first gitch and increasing unti their sum reaches 200fC. Aso, in Figure 2(d, one gate is assumed to be a FO1 and the other one is FO2. As it can be seen from Figure 2, for a FO1 gate, the sum of the gitch sizes exceeds the singe gitch size, when they resut from the same overa coected charge, whie it is smaer than the singe puse size for the FO2 gate for smaer coected charge vaues. For arger coected charge vaues, the curve sum converges to the curve origina, as it can be seen in the figure. Finay, when considering a specific kind of a transient faut, that is, a transient faut induced by a specific event (e.g., cosmic rays, capacitive couping, eectromagnetic interference, etc., it is important to define the range of gitch sizes that can occur due to those events. For exampe, for a radiationinduced transient faut in 130nm technoogy, it has been shown that the duration of a gitch ies in the interva from 30 to 300ps [3], with most gitches having the duration between 100 and 250ps [9]. It has been shown that, among mutipeevent transient induced errors, 90% are the resut of two simutaneous gitches [5], and thus we ony considered this case in our work Faut propagation Soft errors used to be a much greater concern in memories than in ogic circuits, mosty due to the impact of three important masking factors that affect the propagation of a gitch through combinationa circuit [6]: ogica masking occurs if the gitch arrives to the input of a gate when at east one of its other inputs has a controing vaue; eectrica masking can attenuate or even competey mask the gitch that is not arge enough compared to the deay of a gate through which the gitch propagates; atching-window masking occurs when the gitch does not arrive on time at the input of the atch to satisfy its setup and hod time conditions. With technoogy scaing, the impact of these masking factors is decreasing, thus eading to the increased SER in ogic circuits. By taking into account the oint dependence of the three masking factors on circuit topoogy and input vectors, a unified treatment of the three masking factors [6] aso aows for accurate anaysis of reconvergent gitches. Reconvergent gitches occur whenever a puse originating at a given gate in the circuit propagates on more than one path to another gate. In case of a MET, not ony gitches originating from a singe gate, but instead a gitches that resuted from the same partice hit are considered reconvergent. There are severa possibe cases of reconvergent gitches that can occur for both SETs and METs, as shown in [6]. Two gitches that are to be merged may arrive at gate inputs with both controing or both non-controing vaues, or one controing and the other noncontroing vaue. These different cases ead to different output gitches varying in their size and deay, compared to the origina gitches. 4. Proposed transient faut anaysis framework In this section, we describe our approach to modeing and anaysis of SET and MET generation and their propagation through ogic circuits. The pseudo code of our agorithm is given in Figure Faut generation impementation There are severa possibe approaches to the modeing of an SET in terms of the detais incuded in its mode description. For exampe, simpe modes, ike trianguar or trapezoida, incude information about gitch duration and ampitude and possiby about the sope. On the other hand, there are approaches that use more accurate modes, and consequenty need more information about the gitch, ike doubeexponentia current puse [2]. However, there is a tradeoff between the accuracy of the gitch mode and the time needed for a method based on such a mode to estimate the impact of the gitch on circuit outputs. Therefore, in this work we use the former approach that assumes ony gitch duration and ampitude as the gitch parameters and, as it has been shown in [6],[7], aows for the anaysis of circuit soft-error susceptibiity that is within 4% accurate and provides 5000X speedup when compared to detaied HSPICE simuations. A practica impementation for SET modeing was described in [6] where a topoogicay sorted ist of gates for a given circuit is generated first, and then, in one pass through the circuit, a possibe gitches that can occur in the circuit are created and propagated to the primary outputs. In case of a MET, it is aso necessary to determine sets of gates that can be affected by a singe partice hit, that is, gates where gitches determined by a singe MET set originate. Whie in genera the potentia victims of a partice hit can be best determined by 3

4 main { set technoogy parameters; parse input netist; create gate_node_ist; create topoogicay sorted gate ist (sorted_ist; for each gate in sorted_ist { mergegitches(gitch_ist; maskgitches; //ogica and eectrica masking createnewgitch; //new gitches originating at gate sendgitches; //propagate to output neighbors compute error probabiities from fina output ADDs; Figure 3. The proposed agorithm. using the ayout information, at ogic eve, in the absence of ayout information, the two cases described in Section 3.1 (i.e., a gate and its fanin or fanout neighbors, or two gates with common fanin or fanout neighbors may be considered as good candidates for anayzing METs. Thus, our framework takes as inputs the gate-eve description of the circuit and information about fanin and fanout neighbors of each gate and assumes that METs occur at pairs of gates that are neighbors as defined by the two cases above. This does not affect the generaity of our framework, since the ayout information can easiy be incorporated into the input circuit description Faut propagation impementation Since it has been shown in [6] that a unified treatment of the three masking factors (ogica, eectrica and atchingwindow masking is important and most of the previous work anayzed those factors independenty, we appy here the unified symboic mode proposed in [6]. The main idea of the approach proposed in [6] is that the impact of the three masking factors can be modeed using BDDs and ADDs. When duration and ampitude ADDs representing a gitch originating at a given gate G i are created, they are further propagated to the fanout neighbors of gate G i, and there they are modified according to ogica masking, the deay of those gates, and the attenuation mode. This approach can be modified such that it accounts for mutipe gitches occurring as a resut of a singe partice strike. The important advantage of the proposed mode is that it concurrenty computes the propagation and the impact of singe-event transients originating at different interna gates of the circuit. This is made possibe by assigning the originating gate identifier to the duration-ampitude ADD pair associated with each gitch. To be abe to appy the same concurrent computation to mutipe gitch propagation, we need to incude the foowing modifications to the mode. First, instead of assuming the occurrence of ony one gitch at a time, it is necessary to keep track of severa gitches. This requires the specific information of the ist of gates at which gitches occurred to be assigned to a duration-ampitude ADD pairs that correspond to gitches within a given mutipeevent transient set. Thus, when compared to the SET case, the propagation of METs requires more information. The compexity of simpe gitch propagation and attenuation functions is not much affected, due to the fact that the number of gitches that need to be considered at one pass through the circuit is 2N, where N is the number of gitches considered in the singe-event transient case. However, the number of cases that need to be considered when reconvergent gitches are merged increases, as described next. Whenever gitches beonging to the same MET set arrive to the inputs of the same gate, the reconvergent gitch merging mergegitches (gitch_ist{ findreconvergentgitches(gitch_ist; for each reconvergent_set { findquasisensbdds; //mask gitches with other, controing inputs mutuamasking; //gitch carrying inputs can mask one another for each reconvergent_pair { mergepairs(reconvergent_pair; createnewgitch; //some cases resut in a new gitch [6] removezerogitches; //some gitches may be masked [6] findreconvergentgitches (gitch_ist{ for each gitch i from gitch_ist { for each gitch from gitch_ist { if (gitch i and gitch beong to the same MET set then gitch i and gitch are incuded in the same reconvergent_set; mergepairs (reconvergent_pair { merge two gitches according to reconvergent gitch cases; Figure 4. Gitch merging agorithm. agorithm [6] can be appied (Figure 4. The main difference between the SET and MET case, when considering reconvergence, is the fact that, in the case of METs, two gitches to be merged may originate at different gates affected by the same partice hit. A singe gitch, created at some gate inside the circuit, can represent a gitch in different MET sets, as ong as it is not merged with its couped (coming from the same MET set gitch. The information specific for a given MET set of gitches is generated and is not appicabe anymore to other MET sets (that incude a gitch originating from the same gate as the merged gitch. Once gitches are merged, it is necessary to represent the resuting gitch(es as new gitch(es with their specific MET set ists. Therefore, this increases the number of gitches, but at the same time decreases the size of individua gitch MET gate ists Mutipe-event upset probabiity computation To find the probabiity that an MET, representing a set of gitches (1,2,, n, originating at a set of gates (G 1, G 2,, G n is atched at a given output F, a possibe vaues for the duration and ampitude of gitches arriving to the output F are found. For each origina partice hit, there may be severa corresponding gitches that propagated to the output F from different gates. To this end, we define the foowing event: E an event that occurs when any of the gitches originating from one of the gates of the MET set is atched at the output F. More specificay, we can find, for each output F, the probabiity of faiing due to an MET with initia durations d init = (d init,1, d init,2,, d init,n and initia ampitudes a init = (a init,1, a init,2,, a init,n that originated at a given set of gates G i = (G 1, G 2,, G n as: d P(E init,a init i, = P(F fais G i fais gitches = (d init,a init (3 Simiar to [6], we find the probabiity of event E by summing over a possibe gitch durations, D k, that occur at a given output and resut from the propagation of gitches from a given gate MET set: D P(E = k (t setup + t hod P(D = D k (4 k T ck d init where T ck is the cock period, t setup and t hod are the setup and hod time of the atch, respectivey, and d init is the initia duration of the gitch that has duration D k at the output. 4

5 TABLE I Agorithm runtime for severa benchmarks circuits for the three cases: an SET, two simutaneous METs, and two correated MBUs. S27 S208 S298 S444 S526 S1196 S1238 SET MET MBU We can now compute the Mean Error Susceptibiity (MES of a given output F, for a given assumed set of initia gitch durations and ampitudes, (d init, a init, as the average probabiity of output F faiing due to a possibe MET sets that can occur in the circuit, given different input probabiity distributions: MES(F d init,a init = n f n G k=1 i=1 d P(E init,a init i, n G n f (5 where n Gi is the cardinaity of the set of MET gate sets of the circuit, {G i and n f is the cardinaity of the set of probabiity distributions, {f k, associated to the input vector stream. 5. MBU modeing Whie previous discussion focused mainy on the occurrence of transient fauts and their propagation through combinationa part of the circuit, in this section we describe our mode for mutipe fip-fop upsets in sequentia circuits. The main idea here is to determine the impact on the fina computed SER when transient fauts stemming from a singe partice hit affect the state of more than one fip-fop and thus propagate through the circuit as mutipe errors in the cyces foowing the cyce when the hit occurred. The anaysis of sequentia circuits can be spit into two main stages [7]: Stage I, representing the cyce when the hit occurs and Stage II, representing a the foowing cyces. In Stage I, it is necessary to incude the impact of a three masking factors, whie in Stage II, after gitches affected fipfop states, ony ogica masking is considered. Therefore, when fina gitch duration and ampitude ADDs at primary outputs or next state ines are found in Stage I, it is possibe to extract the information about the error correations between different state ines. In other words, it is possibe to find the probabiity of two or more next state ines faiing due to an SET at a given gate (or, an MET at a given set of gates. The computation of conditiona probabiities in Stage II wi assume mutipe errors, which aows us to appy the modeing described in Section 4. The average error probabiity at a given output at Stage II can then be found using the conditiona probabiities computed at Stage II, and mutipe event (i.e., doube, tripe, etc. error probabiities in Stage I: 1 2 k,d P(F init,a init = P(F k 1,d F init,a init 1,d P(F init,a init (6 + P(F k 1,d F init,a init 1,d 1 F init,a init 1,d 2 P(F init,a init 1,d 1 F init,a init P(F k 1,d F init,a init 1,d I i P( F init,a I init i i i 1 2 ns where k,d P(F init,a init is the probabiity of output at the substage k faiing, given an initia gitch duration and ampitude sets, d init and a init. P(F k 1,a F init,d init is the probabiity of error at the output at the stage k, given that an error was atched at the Figure 5. SER for different benchmarks, when two simutaneous METs (MET2, MBU1, two correated MBUs (MET1, MBU2 or an SET are assumed at Stage I. state ine after the first stage with the probabiity of error at state ine given by: P(F 1,d init,a init = n G i=1 1,d P(E init,a init i, n G (7 As described above, we can find the probabiities of mutipe state-ine errors, simiary to the singe error from equation (7 and incude those probabiities in equation (6. 6. Error computation The overa probabiity of output F faiing, P(F, due to different MET sets, with different initia gitch durations and ampitudes and for different input vector probabiity distributions, can be defined using the MES metric. Assuming a uniform distribution of duration-ampitude pairs (d,a aong the surface S = (d max d min (a max a min, for individua gitches, by partitioning the surface of each gitch from the MET set into sub-surfaces (as described in [6], we can find the probabiity P(F as the weighted average of an MES across a combinations of MET eement sub-surfaces: P(F = n n n 1 mn... 1 n 1 n m1 (d MES(F 1,d 2,...,d n,(a m1,a m2,...,a mn d 1 a 1 S n S 1 ( n =1m n =1 1 =1 m 1 =1... d n a n (8 where, for each i=1,..,n: d i = d min + i d i and d max = d min + n i d i a mi = a min + m i a i and a max = a min + n mi a i. Without any oss of generaity, in equation (8, we assume that a MET gitch size combinations have equa probabiity of occurrence. It is, however, straightforward from the equation (8 that different MET probabiities can be easiy incuded. One of the important aspects of the proposed modeing framework is that it is independent of the transient faut source as we as the circuit impementation as ong as the function of the circuit can be described using BDDs and ADDs. Next, the fina ADDs created for individua gate-output pairs are free of any faut-specific information and incude ony information about circuit topoogy and technoogy node parameters. They can further be used to compute output and circuit error probabiities when more specific information about circuit inputs is provided. Finay, the incusion of transient-faut origin specific parameters as externa inputs to the framework aows for computation of error rates. For exampe, in case of radiation-induced soft errors, the SER can be found by using the output error probabiities from equation (8 as: 5

6 Figure 6. Changes in error probabiity in the cyces foowing partice hit for severa benchmarks for 80ps initia gitch, assuming two simutaneous METs (top two charts or two correated MBUs (bottom two charts at Stage I. SER F = P(F R eff R PH A circuit (9 where R PH is the partice hit rate per unit of area, R eff is the fraction of partice hits that resut in charge generation, and A circuit is the tota siicon area of the circuit. As an exampe of how the proposed framework can be appied to transient fauts, we use the soft error specific parameters defined above to compute the error rates, as shown in the next section. 7. Experimenta resuts In this section, we present the resuts obtained using the proposed framework. The technoogy used is 70nm, Berkeey Predictive Technoogy Mode [12]. The cock cyce period (T ck used is 250ps, and setup (t setup and hod (t hod times for the atches are assumed to be 10ps each. V dd is assumed to be 1V. The benchmark circuits are chosen from ISCAS 89 and mcnc 91 suites. The proposed framework is impemented in C++, and run on a 3GHz Pentium 4 workstation running Linux. In TABLE I, we report the runtime of our framework for severa benchmarks, and compare the times for three cases: singe-event transient (SET, two simutaneous transients stemming from a singe hit (MET 2 and two correated bit upsets resuting from a singe faut (MBU 2. We show in Figure 5 the SER resuts for different benchmark circuits, assuming doube gitches and doube fipfop upsets. It is important to note here that the resuts of previous research have shown doube-event transients to be most probabe among a mutipe-event transients (92% of a mutipe transients [5]. Hence, the simutaneous occurrence rate of three or more transients as a resut of a singe partice hit may be assumed negigibe. The SER vaues in the case of two METs are obtained by averaging across severa gitch size combinations (80ps and 60ps, 80ps and 40ps, 60ps and 40ps. The SER vaues for two MBUs are averaged across different initia SET gitch sizes (100ps, 80ps, 60ps and 40ps and the output probabiity vaues are obtained by summing over a MBU pair combinations that can occur in Stage I. Note that these MBU resuts show ony a part of the sum in equation (8 and thus the MBU vaues are smaer than the SET vaues. As can be seen from the presented resuts, the impact of mutipeevent transients (METs and mutipe-state ine errors (MBUs varies across different circuits. For exampe, in case of benchmarks S1196 and S1238, incuding ony the impact of two correated MBUs, resuts in an underestimation of circuit s soft error rate, whie for circuits S526 and S208 it is very cose to the SET case. In Figure 6, we show the changes in average output error probabiity for severa benchmarks in cyces foowing the partice hit, assuming initia gitch duration of 80ps and two simutaneous METs (top two charts or two correated MBUs (bottom two charts in Stage I. The resuts again show that different circuits behave differenty with respect to mutipe fauts or mutipe fip-fop upsets. The probabiity of error at the output, in the cyces after the partice hit, can foow a three trends: decrease rapidy, remain at about the same eve, or increase. 8. Concusion In this work, a probabiistic symboic modeing methodoogy for efficient and accurate estimation of the susceptibiity of ogic circuits to transient fauts is proposed. The main idea behind the proposed work is to aow for the anaysis of the susceptibiity of individua outputs to errors stemming from singe and mutipe transient fauts. 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