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All these systems consist of \ parallel processes which communicate with each other, exchanging data or \ combining results from the different processes. An area of concern is the \ reliability and fault tolerance of these systems [Somai and Vaidya 1997]. The \ most difficult types of failure to handle are transient failures caused by \ environmental factors or design errors, as traces of their nature are long \ gone by the time they are detected. To overcome this difficulty, a number of \ software techniques have been suggested. The most common technique involves \ some form of rollback and recovery method [Wu et al. 1990, Chandy et al. \ 1975, Kant 1978, Gray and Siewiorek 1991]. Typically, the state of the system \ is saved at periodic intervals, a process called ", StyleBox["checkpointing", FontWeight->"Bold"], ". One of the popular error detection techniques is the use of acceptance \ tests, for example a check for proper variable type (e.g. a non-negative \ value is expected, but a negative value is given as a function input). Upon \ detection of an error, the system needs to recover, that is to start over \ from a checkpoint (= saved system state) that was established before the \ failure occurred. The main task for any recovery method consists of \ identifying a previously established checkpoint from which recovery is \ possible.\n\nThe ", StyleBox["Mathematica ", FontSlant->"Italic"], "simulation described in this article compares two different schemes, the \ iterative and the selective rollback methods, for identifying the checkpoint \ from which recovery is to be attempted. The first method proceeds one step at \ a time, rolling back to the next most recent checkpoint and then attempting \ recovery. The second method selects a checkpoint based on the distribution \ function of the latency times (= time between a failure and its detection). \ We simulate a set of ", StyleBox["n", FontSlant->"Italic"], " parallel processes to collect \"data\" on the latency distribution and to \ compare the cost of recovery for the two schemes. Furthermore, a \ visualization of the processes will illustrate the sequence of \ inter-connections, failures and acceptance tests." }], "Text"], Cell[CellGroupData[{ Cell["Model Description", "Section", CellDingbat->None], Cell[TextData[{ "Before formalizing the model, we give an example. Below is a system with 4 \ processes ", Cell[BoxData[ \(TraditionalForm\`P\_1\)]], ",", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(P\_2\)\)\)]], " , ", Cell[BoxData[ \(TraditionalForm\`P\_3\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`P\_4\)]], ". Checkpoints ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "4"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "5"], TraditionalForm]]], ", and ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "6"], TraditionalForm]]], " indicate saved system states. Arrows between processes indicate \ inter-connections. The small rectangle (\"fail\") indicates the occurrence of \ a transient error in process 3. Through the inter-connections, both process \ 2 (directly) and process 1 (indirectly via process 2) may have received \ faulty data. Only process 4 is not affected. Once the acceptance test (AT) in \ process 1 detects the error, all processes need to be rolled back to \ checkpoint ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "4"], TraditionalForm]]], " to ensure recovery." }], "Text"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .24923 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.126374 0.0586081 0.00593407 0.0586081 [ [ 0 0 0 0 ] [ 1 .24923 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .24923 L 0 .24923 L closepath clip newpath % Start of user PostScript /mathtops { gsave MBeginOrig moveto MEndOrig currentpoint grestore } bind def /MAtocoords { mathtops 4 2 roll mathtops 4 copy pop pop 3 -1 roll sub /arry exch def exch sub /arrx exch def arrx dup mul arry dup mul add sqrt /arrl exch def translate } bind def /MAarrowhead1 { gsave MAtocoords arrl 0. eq { 0 0 Mdot } { currentlinewidth 1 Mabswid currentlinewidth dup scale setlinewidth [ arrx arrl div arry arrl div -1 arry mul arrl div arrx arrl div 0 0 ] concat -15. 3.75 moveto 0 0 lineto -15. -3.75 lineto fill -15. 3.75 moveto 0 0 lineto -15. -3.75 lineto -15. 3.75 lineto stroke } ifelse grestore } def % End of user PostScript 0 g .5 Mabswid .05018 .06454 m .97619 .06454 L s .05018 .12315 m .97619 .12315 L s .05018 .18176 m .97619 .18176 L s .05018 .24037 m .97619 .24037 L s .12051 .06161 m .12051 .2433 L .13223 .2433 L .13223 .06161 L F .41355 .06161 m .41355 .2433 L .42527 .2433 L .42527 .06161 L F .70659 .06161 m .70659 .2433 L .71832 .2433 L .71832 .06161 L F .29634 .1759 m .29634 .18762 L .30806 .18762 L .30806 .1759 L F .27289 .06454 m .29048 .12315 L s % Start of user PostScript 2.5 1. 2.8 2. 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which schedules (global) checkpointing and initiates recovery (this latter \ process does not show up in the simulation). In each of the ", StyleBox["n ", FontSlant->"Italic"], "processes, inter-connections between the processes, failures and \ acceptance tests occur at random times. \nTo model this system, we assume \ that the waiting times between events (inter-connections, failures, and \ acceptance tests) are all independent exponential random variables. A failure \ in process ", StyleBox["i ", FontSlant->"Italic"], "is either detected by an acceptance test in the same process or by an \ acceptance test in process ", StyleBox["k", FontSlant->"Italic"], " if there was a direct or indirect connection between process ", StyleBox["i", FontSlant->"Italic"], " and process ", StyleBox["k ", FontSlant->"Italic"], "after the failure occurred. We will use the following parameters for the \ exponential waiting times:\n\t", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_ij\)]], " for inter-connections between process ", StyleBox["i", FontSlant->"Italic"], " and process ", StyleBox["j", FontSlant->"Italic"], "\n\t", Cell[BoxData[ \(TraditionalForm\`\[CurlyPhi]\_i\)]], " for failures in process ", StyleBox["i\n\t", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_i\)]], " for acceptance tests in process ", StyleBox["i.\n", FontSlant->"Italic"], "We assume that checkpointing occurs at fixed intervals of length ", StyleBox["C ", FontSlant->"Italic"], "and that the time it takes to either save or reload a system state, \ denoted by ", StyleBox["CL", FontSlant->"Italic"], ", is also constant. Checkpoints are numbered from 1 to ", StyleBox["tot+1", FontSlant->"Italic"], ", where ", StyleBox["tot ", FontSlant->"Italic"], "= # of checkpointing intervals (= period between two checkpoints) covered \ so far. ", StyleBox[" ", FontSlant->"Italic"], "Due to limitations of storage space, only a certain number of checkpoints \ can be maintained at any one time. Thus, once this number ", StyleBox["m", FontSlant->"Italic"], " is reached, the oldest checkpoint is overwritten by a new checkpoint. " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Simulation", "Section", CellDingbat->None], Cell[TextData[{ "The user provides the following parameters: \n1) ", StyleBox["n", FontWeight->"Bold"], ", the number of processes (not counting the control process); \n2)", StyleBox[" icrange", FontWeight->"Bold"], ", the range (in hours) for average times between inter-connections;\n3)", StyleBox[" failrange", FontWeight->"Bold"], ", the range (in hours) for average times between failures; and\n4) ", StyleBox["acceptrange", FontWeight->"Bold"], ", the range (in hours) for average times between acceptance tests." }], "Text"], Cell["\<\ From the literature [ Shin and Lee 1984, Pamula 1990], the following \ intervals for average times (in hours) between events were taken: \tInter-connections \t[ 0.25, 1] \tFailures\t\t[ 10, 40 ] \tAcceptance Tests\t[ 1, 2 ] \ \>", "Text"], Cell["Thus, for an example with 4 processes, we enter:", "Text"], Cell["\<\ n = 4; icrange = {.25,1}; failrange = {10,40}; acceptrange = {1,2}; \ \>", "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[{ "\nSince we are assuming exponential random variables, the parameter for \ the distribution is the reciprocal of the expected value. Thus, to determine \ the individual parameters, the range intervals are transformed by computing \ the intervals for the reciprocals. Then a random number from these new \ intervals is selected for each event. This is done using the function ", StyleBox["CreateRates", FontWeight->"Bold"], ". This function has as its arguments the intervals for the average times. \ The rates for each type of event are created as the lists ", StyleBox["icrates, failrates, ", FontSlant->"Italic"], "and", StyleBox[" acceptrates", FontSlant->"Italic"], ". The different event rates are combined into an ", StyleBox["n", FontSlant->"Italic"], "-by-(", StyleBox["n+1", FontSlant->"Italic"], ") matrix. The inter-connection rate ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[Lambda]\_ij\), FontSlant->"Italic"], " "}], TraditionalForm]]], "between process ", StyleBox["i ", FontSlant->"Italic"], " and process ", StyleBox["j", FontSlant->"Italic"], " is listed as the (", StyleBox["i,j", FontSlant->"Italic"], ")", StyleBox[" ", FontSlant->"Italic"], "entry, the failure rate ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyPhi]", StyleBox["i", FontSlant->"Italic"]], TraditionalForm]]], " for process ", StyleBox["i", FontSlant->"Italic"], " is stored as the (", StyleBox["i,i)", FontSlant->"Italic"], " entry and the acceptance rate", Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox["\[Alpha]", StyleBox["i", FontSlant->"Italic"]]}], TraditionalForm]]], " for process ", StyleBox["i", FontSlant->"Italic"], " is stored as the (", StyleBox["i", FontSlant->"Italic"], ", ", StyleBox["n+1)", FontSlant->"Italic"], " entry." }], "Text"], Cell[BoxData[ \(CreateRates[ic_List, fails_List, accept_List] := \n\t Module[\n\t\t\t{icrates = Table[Random[Real, {1/ic[\([2]\)], 1/ic[\([1]\)]}], {i, n}, {j, n - 1}], \n\t\t\tfailrates\ = Table[Random[Real, {1/fails[\([2]\)], 1/fails[\([1]\)]}], {i, n}], \ acceptrates = Table[Random[Real, {1/accept[\([2]\)], 1/accept[\([1]\)]}], {i, n}], \n\t\t\trates\ = \ Table[0, {n}]}, \n\n\t\tDo[ rates[\([i]\)]\ = \ \n\t\t\t\t\t\tAppend[ Insert[icrates[\([i]\)], failrates[\([i]\)], i], \n\t\t\t\t\t\t\t\t\t\t\tacceptrates[\([i]\)]], {i, 1, n}]; \n\t\trates]\)], "Input", InitializationCell->True], Cell[TextData[{ "Here is an example of the parameters resulting from the function ", StyleBox["CreateRates", FontWeight->"Bold"], ":\n\n\t\t\t", Cell[BoxData[GridBox[{ {\(\[Lambda]\_ij\), "1", "2", "3", "4"}, {"1", "*", "3.50163", "1.85276", "3.91859"}, {"2", "1.83752", "*", "2.3015", "3.29872"}, {"3", "3.34154", "1.15867", "*", "2.80303"}, {"4", "3.76839", "3.04845", "1.42542", "*"} }, RowLines->{0.5, False}, ColumnLines->{0.5, False}]]], "\n\t\t\t\n\t\t\t ", Cell[BoxData[GridBox[{ { StyleBox["i", FontSlant->"Italic"], \(\[CurlyPhi]\_i\), StyleBox[\(\[Alpha]\_i\), FontSlant->"Plain"]}, { StyleBox["1", FontSlant->"Plain"], StyleBox["0.0803972", FontSlant->"Plain"], StyleBox["0.0871607", FontSlant->"Plain"]}, { StyleBox["2", FontSlant->"Plain"], StyleBox["0.0979104", FontSlant->"Plain"], StyleBox["0.0949468", FontSlant->"Plain"]}, {"3", "0.0801946", "0.970564"}, {"4", "0.0657209", "0.770418"} }, RowLines->{0.5, False}]], GridBoxOptions->{ColumnLines->{0.5, False}}], "\n\t\t\t " }], "Text"], Cell["\<\ Once all the rate parameters are computed, we need to determine the time until the next event. This time is the minimum of exponential times, hence also an exponential time with parameter \ \>", "Text"], Cell[BoxData[ RowBox[{"\t\t\t", RowBox[{"\[Mu]", " ", "=", " ", RowBox[{ FormBox[\(\[Sum]\+\(i = 1\)\%n\(\[Sum]\+\(j \[NotEqual] i\)\%n \ \[Lambda]\_ij\)\), "TraditionalForm"], "+", FormBox[\(\[Sum]\+\(i = 1\)\%n \[CurlyPhi]\_i\), "TraditionalForm"], "+", FormBox[\(\[Sum]\+\(i = 1\)\%n \[Alpha]\_i\), "TraditionalForm"]}]}]}]], "Input"], Cell[TextData[{ "Furthermore, the probability that the next event is the event with \ parameter \[Rho]\[Element]{", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_ij\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[CurlyPhi]\_i\)]], ",", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_i\)]], "} is given by \[Rho]/\[Mu]. These probabilities are used as input for the \ function ", StyleBox["WhichOne", FontWeight->"Bold"], " to determine which event takes place next. Imagine the interval [0,1] \ partitioned by the event probabilities (they sum to 1). Each segment created \ by the partition can be associated with a specific event. Now a uniform \ random number from [0,1] is selected. This number falls into one of the \ segments, indicating that the corresponding event takes place. The function \ WhichOne makes use of the structure in which the rates, and hence the \ corresponding probabilities, were ordered in the function CreateRates. The \ variables", StyleBox[" to", FontSlant->"Italic"], " and ", StyleBox["from", FontSlant->"Italic"], " are used to keep track of which kind of connections have taken place. If \ ", StyleBox[" to ", FontSlant->"Italic"], " = ", StyleBox[" from, ", FontSlant->"Italic"], " a failure has occurred, and if ", StyleBox["to = n+1", FontSlant->"Italic"], ", then the event is an acceptance test. " }], "Text"], Cell["\<\ WhichOne[probabilities_] := \tModule[{event, eventcheck = 0,m = 0, problist}, \t\tevent = Random[]; \t\t \t\tproblist = Flatten[probabilities]; \t\tWhile[eventcheck < event, \t\t\tm = m+1; \t\t\teventcheck = eventcheck + problist[[m]] \t\t\t]; \t\tIf[Mod[m,n+1]==0, \t\t\tto = n+1; from = Quotient[m,n+1], \t\t\tto = Mod[m,n+1];from = Quotient[m,n+1]+1; \t\t\t] \t\t] \ \>", "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[{ "To simulate the system of parallel processes and the detection of errors, \ we need to keep track of when and where the failures occurred and which \ processes have had inter-connections. This history will allow us to determine \ which failures can be detected once an acceptance test occurs. Failures and \ inter-connections are recorded, whereas acceptance tests either result in \ detection of an error and corresponding reset, or have no consequence at all. \ The ", StyleBox["n", FontSlant->"Italic"], "-by-", StyleBox["n", FontSlant->"Italic"], " matrix", StyleBox[" ", FontWeight->"Bold"], StyleBox["descendants", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Bold"], "keeps track of inter-connections and the list ", StyleBox["failures", FontSlant->"Italic"], " keeps tracks of whether or not a failure has occurred. Let's look at the \ different events and what effects they have. " }], "Text"], Cell[CellGroupData[{ Cell["Failure occurred ", "Subsubsection", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{20, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "If the failure is the very first one to occur in the system, then ", StyleBox["every", FontVariations->{"Underline"->True}], " element in the matrix ", StyleBox["descendants", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Bold"], "is set to 0 as none of the previous connections are of any relevance to \ future failure detection. If the failure occurring is the first at a \ particular process, then any previous connections of this process to other \ processes are erased by setting the corresponding elements of the matrix ", StyleBox["descendants ", FontSlant->"Italic"], "to zero. In both cases, the relevant entry in ", StyleBox["failures", FontSlant->"Italic"], " is set to 1 to record the error. If the simulation is used for data \ collection, then the time at which the failure occurred is recorded in the \ list ", StyleBox["failuretimes", FontSlant->"Italic"], "." }], "Text", CellDingbat->None, CellMargins->{{20, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Acceptance Test (AT) occurred ", "Subsubsection", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{20, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "If no failure has occurred in the process at which the AT takes place, \ then all connections from this process to others are removed by setting the \ corresponding entries in ", StyleBox["descendants", FontSlant->"Italic"], " to zero (no faulty data has been propagated). For any process in which a \ failure is detected, all previous connections are erased and the entry in ", StyleBox["failures", FontSlant->"Italic"], " is set to zero. If the simulation is used for data collection, then the \ latency time is recorded and ", StyleBox["failuretime", FontSlant->"Italic"], " is reset to zero; if the simulation is used for cost comparison, then \ error recovery is initiated." }], "Text", CellDingbat->None, CellMargins->{{20, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Inter-Connection occurred ", "Subsubsection", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{20, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The connection is recorded by setting the corresponding element in ", StyleBox["descendants ", FontSlant->"Italic"], "to one. Any resulting indirect inter-connections are recorded in the same \ manner." }], "Text", CellDingbat->None, CellMargins->{{20, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Data Collection", "Section", CellDingbat->None], Cell[TextData[{ "We are now ready for the first part of the simulation, namely collecting \ data for the latency distribution. A total of five simulations, each up to \ time 200 (hours), are executed. In each of the simulations, the rate \ parameters will be different, reflecting the changing nature of transient \ errors. The latency times are written to the file simlagtimes. The time clock \ is advanced by exponential times using the ", StyleBox["Mathematica", FontSlant->"Italic"], " package Statistics`ContinuousDistributions`. Note that the time clock \ does not take into account the checkpointing period, as we assume that no \ errors occur during those times. The code for the program is contained in the \ notebook ", StyleBox["Simdata.nb", FontFamily->"Courier"], "." }], "Text"], Cell[TextData[{ "To visualize the latency data, we can use the ", StyleBox["Mathematica", FontSlant->"Italic"], " packages Statistics`DataManipulation` and Graphics`Graphics`." }], "Text"], Cell[BoxData[{ \(<< Statistics`DataManipulation`\), \(<< Graphics`Graphics`\)}], "Input"], Cell[BoxData[ \(\(lagtime\ = \ ReadList["\"];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(dx = 0.2;\)\), "\n", \(\t\ \ \ \(freq\ = \ BinCounts[lagtime, {0, Max[lagtime], dx}];\)\), "\n", \(\ \ \ \ \ \ \(midpoints\ = \ Range[dx/2, Max[lagtime] + dx/2, dx];\)\), "\n", \(\ \ \ \ \ \ BarChart[\n\t\t\ \ \ \ \ \ \ \ \ Transpose[{freq, midpoints}], PlotRange -> All, \n\t\t\t\t\t\tPlotLabel\ -> FontForm["\", \ {"\", 12}]]\n\)}], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix 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0.463697}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[BoxData[ \(\n\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Visualization of the Simulation", "Section", CellDingbat->None], Cell[TextData[{ "We visualize the interplay of the processes using an ", StyleBox["n", FontSlant->"Italic"], "-by-", StyleBox["n", FontSlant->"Italic"], " square grid together with a single column. An inter-connection from \ process ", StyleBox["i", FontSlant->"Italic"], " to process", StyleBox[" j", FontSlant->"Italic"], " is shown by coloring the square whose left lower corner is at position ", StyleBox["(i,j)", FontSlant->"Italic"], ". Different colors are used for direct (cyan) and indirect (magenta) \ connections. A failure in process ", StyleBox["i", FontSlant->"Italic"], " is shown as black square on the diagonal of the grid in the (", StyleBox["i,i", FontSlant->"Italic"], ") position. An acceptance test in process ", StyleBox["i", FontSlant->"Italic"], " is shown in row ", StyleBox["i", FontSlant->"Italic"], " of the additional column (counting upwards). The graphical display is \ updated in parallel with the lists ", StyleBox["failure", FontSlant->"Italic"], " and ", StyleBox["descendants", FontSlant->"Italic"], ". Setting an element to zero in either of these two lists is reflected by \ showing the corresponding square empty. Acceptance tests are erased in the \ next step, as they do not have any relevance to future failure detection. \n\n\ Three functions contribute to the visual display: ", StyleBox["grid, Conn", FontWeight->"Bold"], ", and ", StyleBox["DropConn", FontWeight->"Bold"], ". The function ", StyleBox["grid", FontWeight->"Bold"], " produces the basic grid for the start-up. The ", StyleBox["n", FontSlant->"Italic"], "-by-", StyleBox["n", FontSlant->"Italic"], " grid shows connections and failures, whereas the n-by-1 column shows the \ acceptance tests." }], "Text"], Cell["\<\ (*Joe Single-space next code and below, please. Thanks, David *)\ \>", "Text"], Cell["\<\ grid[n_] := Module[{dim = n, i}, \tGraphics[ \t{Table[{Line[{{i,0},{i,dim}}], \t\t\tLine[{{0,i},{dim,i}}]},{i,0,dim}], \t Table[Line[{{dim+i,0},{dim+i,dim}}],{i,1,2}], \t Table[Line[{{dim+1,i},{dim+2,i}}],{i,0,dim}]},\t\t \t AspectRatio -> Automatic]] \t \ \>", "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[{ "The function ", StyleBox["Conn", FontWeight->"Bold"], " selects the color depending on the event, as well as the text to be \ displayed. It returns a colored square at the appropriate position, which is \ then added to the current display." }], "Text"], Cell["\<\ Conn[x_, y_, c_] := Module[{col,txt,i,j}, col = Which[c == 0, RGBColor[0,1,0], (*Acceptance*) \t\t \t c == 1, RGBColor[0,.7,.8], (*direct connection*) \t\t \t c == 2, RGBColor[.8,.2,.5], (*indirect connection*) \t\t \t c == 3, RGBColor[0,0,0]]; (*failure*) \tj = Which[y == n+1,\tx, \t\t\t y != n+1, y]; \ti = Which[y == n+1,\tn+2, \t\t\t y != n+1, x];\t\t \t \ttxt = Which[y == n+1, StringForm[\"`1`\",from], \t\t\t\tx == y, StringForm[\"\"], \t\t\t\ty != n+1, StringForm[\"`1`->`2`\",i,j]]; \t\t\t\t\t \t \tGraphics[{{col, Rectangle[{i-1,j-1},{i,j}]}, \t\t\t Line[{{i,j},{i-1,j},{i-1,j-1},{i,j-1}, \t\t\t {i,j}}], \t\t\t {RGBColor[0,0,0], \t\t\t Text[txt,{i-.5,j-.5}]}}, \t\t\t AspectRatio -> Automatic] ]\t \ \>", "Input", InitializationCell->True], Cell[TextData[{ "The function ", StyleBox["DropConn", FontWeight->"Bold"], " handles the resetting aspect once an acceptance test takes place. Rather \ than just adding empty squares to the Graphics display to overwrite displays \ of earlier events, this function selectively removes the appropriate colored \ squares. The input to the function ", StyleBox["DropConn", FontWeight->"Bold"], " is the process number, the current list of Graphics elements for display, \ and the variable ", StyleBox["case", FontSlant->"Italic"], ". If ", StyleBox["case", FontSlant->"Italic"], " == False, then only connections are removed. If ", StyleBox["case", FontSlant->"Italic"], " == True, then acceptance tests and failures will also be removed. The \ function ", StyleBox["DropConn ", FontWeight->"Bold"], "computes the positions of the Graphics elements to be removed, and then \ uses the built-in function Delete to complete the task." }], "Text"], Cell[BoxData[ \(\(DropConn[k_, connlist_List, case_] := \n\t Module[{del\ = {}, i, m}, \n\t\t\ \ If[ case\ == \ False, \n\t\t\t\t\t\tDo[ AppendTo[del, Position[connlist, Conn[k, i, m]]], \n\t\t\t\t\t\t\t\t{i, 1, n}, {m, 1, 2}], \n\t\t\ \ \ \ \ \ \ \ \tDo[ AppendTo[del, Position[connlist, Conn[k, i, m]]], \n\t\t\t\t\t\t\t\t{i, 1, n + 1}, {m, 0, 3}]\n\t\t\ \ \ \ ]; \n\ \ \ \ \ \ \ \ \ del\ = \ Flatten[del, 1]; \n\t\t\t\tDelete[connlist, del]\n\t];\)\)], "Input", InitializationCell->True], Cell[TextData[{ "The file Simviz.nb contains the code for the visualization. Whenever the \ lists ", StyleBox["descendants", FontSlant->"Italic"], " and ", StyleBox["failure", FontSlant->"Italic"], " are updated, the corresponding visual elements are added or removed. \ Failure times are not tracked and only a single simulation is executed. Here \ are some examples of output from the file Simviz.nb. Note that for each \ event, the cumulative relevant history is displayed. Events are numbered \ consecutively, and the current event is indicated in the title of the graph. \ In all three cases, two consecutive events are displayed to show the \ dynamics.\n" }], "Text"], Cell[CellGroupData[{ Cell["Indirect Connections", "Subsubsection", CellDingbat->"\[FilledSmallCircle]"], Cell[TextData[{ "The first grid shows the relevant history up to Event 10, an \ inter-connection from process 1 to process 4. Event 11 is an \ inter-connection from process 4 to process 2, displayed as a cyan square in \ row 2, column 4 with label ", StyleBox["4\[Rule]2", FontSlant->"Italic"], ". 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The system needs \ to roll back at least to the checkpoint just prior to the failure to achieve \ recovery (we assume that no other errors occur during the recovery phase). \ The easiest way to achieve recovery is the ", StyleBox["iterative method", FontWeight->"Bold"], ". The system is rolled back to the most recent checkpoint and starts over \ from there. 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>?oo00<00?ooool00ooo00<00?ooool05_oo00<00?ooool05ooo00<00?ooool00ooo00<00?ooool0 `_oo00<00?ooool06?oo00<00?ooool00ooo00<00?ooool02?oo0`00;ooo000hool01000ooooo`00 0_oo00<00?ooool05_oo00@00?ooool001Koo`04003ooooo0002ool00`00ooooo`32ool01000oooo o`005ooo00@00?ooool000;oo`03003ooooo00[oo`03003ooooo02goo`00>Ooo0`0000Coo`000000 01Soo`<001Ooo`<00004ool000000034ool3000Hool300001?oo000000002_oo0P00;ooo003ooonK ool00?oooi_oo`00ooooVooo003ooonKool00?oooi_oo`00ooooVooo003ooonKool00001\ \>"], ImageRangeCache->{{{82.875, 491.875}, {428.3125, 370.8125}} -> {-1.49798, 6.94911, 0.017392, 0.017392}}], Cell[TextData[{ "The cost for recovery from checkpoint ", StyleBox["k", FontSlant->"Italic"], " is given by \n\n\t", StyleBox["T(k)=(tot+1 - k) CT + CL + d\n\t", FontSlant->"Italic"], "\nLet's now compare the two algorithms and see how they arrive at the \ total cost.\n" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Total Cost of Recovery for the Iterative Method", "Subsection"], Cell[TextData[{ "Assume that the error occurred between ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "k"], TraditionalForm]]], StyleBox[" ", FontSlant->"Italic"], " and ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], \(k - 1\)], TraditionalForm]]], ". The iterative method first rolls back to ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], RowBox[{ StyleBox["tot", FontSlant->"Italic"], "+", "1"}]], TraditionalForm]]], ", incurring cost ", StyleBox["T(tot+1)", FontSlant->"Italic"], ". Then it rolls back to ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], StyleBox["tot", FontSlant->"Italic"]], TraditionalForm]]], ", incurring cost ", StyleBox["T(tot)", FontSlant->"Italic"], ". This continues until the process has rolled back to ", Cell[BoxData[ FormBox[ FormBox[\(CP\_k\), "TraditionalForm"], TraditionalForm]]], ". The cumulative cost of recovery is thus given by\n\n\t ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox[\(\[Sum]\+\(i = k\)\%\(tot + 1\)\), FontSlant->"Italic"], \(T(i)\)}], " "}], TraditionalForm]]], "=", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ StyleBox[\(\[Sum]\+\(i = 0\)\%rsteps\), FontSlant->"Italic"], RowBox[{"(", \(i\ CT\ + \ CL + d\), " ", StyleBox[")", FontSlant->"Italic"]}]}]}], TraditionalForm]]], " = (", StyleBox["rsteps + 1", FontSlant->"Italic"], ")", Cell[BoxData[ FormBox[ RowBox[{"(", StyleBox[\(\(rsteps\ CT\)\/2 + \ CL\ + \ d\), FontSlant->"Italic"], " ", ")"}], TraditionalForm]]], "\n\t\nwhere", StyleBox[" rsteps", FontSlant->"Italic"], " = ", StyleBox["tot + 1 - k ", FontSlant->"Italic"], "= # of intervals covered by the rollback. The function itrecovery computes \ the cost of iterative recovery based on the time of failure occurrence. The \ variable ", StyleBox["tslcp", FontSlant->"Italic"], " indicates the ", StyleBox["t", FontWeight->"Bold"], "ime ", StyleBox["s", FontWeight->"Bold"], "ince the ", StyleBox["l", FontWeight->"Bold"], "ast ", StyleBox["c", FontWeight->"Bold"], "heck", StyleBox["p", FontWeight->"Bold"], "oint, ", StyleBox[" tcp", FontSlant->"Italic"], "1 denotes the time at which ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "1"], TraditionalForm]]], " was established, and ", StyleBox["cint ", "Input", FontWeight->"Plain"], "denotes ", StyleBox["C", FontSlant->"Italic"], ", the time between checkpoints.\n" }], "Text"], Cell[BoxData[ \(\(\(itrecovery[ failuretime_]\)\(:=\)\(\n\)\(\t\)\(Module[{rsteps}, \n\t\tIf[ failuretime >= \ tcp1, \n\t\t\t\t\trsteps\ = \ Ceiling[\((time - tslcp - failuretime)\)/ cint], \n\t\t\t\ \ \ \ rsteps\ = \ total]; \n\t\t\((rsteps + 1)\) \((rsteps*cycle/2\ + \ CL\ + \ tslcp)\)]\)\(\n\)\(\t\t\t\t\)\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Total Cost of Recovery for the Selective Method", "Subsection"], Cell[TextData[{ "In the iterative method, the selection of the checkpoint from which \ recovery is attempted is based on pairwise comparison of expected cost. If \ the expected cost of recovery from ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "k"], TraditionalForm]]], StyleBox[" ", FontSlant->"Italic"], "is smaller than the expected cost of recovery from ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], \(k - 1\)], TraditionalForm]]], StyleBox[" , ", FontSlant->"Italic"], "then recovery is attempted from ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "k"], TraditionalForm]]], StyleBox["; ", FontSlant->"Italic"], "otherwise, the expected cost of recovery from ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], \(k - 1\)], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], \(k - 2\)], TraditionalForm]]], StyleBox[" ", FontSlant->"Italic"], "are compared (until either recovery is attempted from the checkpoint with \ lower cost or ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "1"], TraditionalForm]]], StyleBox[" ", FontSlant->"Italic"], "is reached).\nIn order to compute the expected cost of recovery from a \ given checkpoint, we need the probabilities that the failure occurred after", Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "k"]}], TraditionalForm]]], ". This probability can be expressed in terms of the latency times (taking \ into account that the exponential time clock for events does not take into \ account saving of the system state):\n\t\n ", StyleBox["P(k):=P(failure occurred after ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "k"], TraditionalForm]]], StyleBox[") = P(lagtime < (tot+1-k) C + d) = EDF((tot+1-k) C + d)", FontSlant->"Italic"], ".\n\nWhen computing the expected cost of recovery from a given checkpoint \ ", StyleBox["k", FontSlant->"Italic"], ", we distinguish as to when the failure has happened. If the failure \ occurred after ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "k"], TraditionalForm]]], ", then rolling back to ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "k"], TraditionalForm]]], " will ensure recovery at cost ", StyleBox["T(k)", FontSlant->"Italic"], ". However, if failure occurred between checkpoints ", StyleBox["m", FontSlant->"Italic"], " and ", StyleBox["m+1 (m"Italic"], ", then rollback from ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "k"], TraditionalForm]]], StyleBox[" ", FontSlant->"Italic"], "will not result in recovery. It is assumed that the rollback proceeds \ iteratively after this first attempt (for the cost computation). Therefore, \ we have the following cases:\n\n ", Cell[BoxData[ StyleBox[GridBox[{ { StyleBox[\(\(Failure\)\(\ \)\(occured\)\(\ \)\), FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[\(with\ probability\), FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[\(at\ cost\), FontFamily->"Times New Roman", FontWeight->"Bold"]}, { RowBox[{ StyleBox["after", FontFamily->"Tahoma", FontSize->10], StyleBox[ RowBox[{ StyleBox[" ", FontFamily->"Tahoma", FontSize->10], StyleBox[" ", FontFamily->"Tahoma", FontSize->10, FontSlant->"Plain"]}]], StyleBox[" ", FontFamily->"Tahoma", FontSize->10, FontSlant->"Plain"], FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "k"], "TraditionalForm"]}], StyleBox[\(P \((k)\)\), FontFamily->"Tahoma", FontSize->10, FontSlant->"Italic"], StyleBox[\(T \((k)\)\), FontFamily->"Tahoma", FontSize->10, FontSlant->"Italic"]}, { RowBox[{ StyleBox["between", FontFamily->"Tahoma", FontSize->10], StyleBox[ RowBox[{ StyleBox[" ", FontFamily->"Tahoma", FontSize->10], StyleBox[" ", FontFamily->"Tahoma", FontSize->10, FontSlant->"Plain"]}]], FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], \(k - 1\)], "TraditionalForm"], StyleBox[" ", FontFamily->"Tahoma", FontSize->10], StyleBox[ FormBox[ RowBox[{ StyleBox[" ", FontFamily->"Tahoma", FontSize->10, FontSlant->"Plain"], FormBox["", "TraditionalForm"]}], "TraditionalForm"], FontFamily->"Tahoma", FontSize->10], StyleBox[" ", FontFamily->"Tahoma", FontSize->10, FontSlant->"Italic"], StyleBox["and", FontFamily->"Tahoma", FontSize->10, FontSlant->"Plain"], StyleBox[" ", FontFamily->"Tahoma", FontSize->10, FontSlant->"Plain"], FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "k"], "TraditionalForm"]}], StyleBox[\(P \((k - 1)\) - P \((k)\)\), FontFamily->"Tahoma", FontSize->10, FontSlant->"Italic"], StyleBox[\(T \((k)\) + T \((k - 1)\)\), FontFamily->"Tahoma", FontSize->10, FontSlant->"Italic"]}, { StyleBox["\[VerticalEllipsis]", FontFamily->"Tahoma", FontSize->10], StyleBox["\[VerticalEllipsis]", FontFamily->"Tahoma", FontSize->10], StyleBox["\[VerticalEllipsis]", FontFamily->"Tahoma", FontSize->10]}, { RowBox[{ StyleBox["between", FontFamily->"Tahoma", FontSize->10], StyleBox[" ", FontFamily->"Tahoma", FontSize->10, FontSlant->"Plain"], FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "1"], "TraditionalForm"], StyleBox[" ", FontFamily->"Tahoma", FontSize->10, FontSlant->"Italic"], StyleBox["and", FontFamily->"Tahoma", FontSize->10, FontSlant->"Plain"], StyleBox[" ", FontFamily->"Tahoma", FontSize->10, FontSlant->"Plain"], FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "2"], "TraditionalForm"]}], StyleBox[\(P \((1)\) - P \((2)\)\), FontFamily->"Tahoma", FontSize->10, FontSlant->"Italic"], StyleBox[\(\(T \((k)\) + T \((k - 1)\) + ... \) + T \((1)\)\), FontFamily->"Tahoma", FontSize->10, FontSlant->"Italic"]} }, GridFrame->True, RowLines->True, ColumnLines->True], FontFamily->"Times New Roman"]]], "\n \nThe expected cost of recovery from ", Cell[BoxData[ \(TraditionalForm\`CP\_k\)], FontSlant->"Italic"], " is thus:\n\n\t", StyleBox["EC(k) = T(k)\[CenterDot]P(k) + \ (P(k-1)-P(k))\[CenterDot](T(k)+T(k-1)) + \[CenterEllipsis]+(P(1)-P(2))\ \[CenterDot](T(k)+\[CenterEllipsis]+T(1))\n\t = \ T(k)\[CenterDot]P(1) + T(k-1)\[CenterDot](P(1)-P(k)) + \[CenterEllipsis]+ \ T(1)\[CenterDot](P(1)-P(2))\n\t =T(k)\[CenterDot]P(1) + ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(r = 1\)\%\(k - 1\)\), StyleBox[\(\(T(r)\)\[CenterDot]\((P(1) - P(r + 1))\)\), FontSlant->"Italic"]}], TraditionalForm]]], StyleBox[" \n\t \n", FontSlant->"Italic"], "This is the setup for the initial selection of the checkpoint for \ recovery. Once rollback has been attempted, two possibilities can occur: \ Recovery was successful (and we are done) or a new checkpoint must be \ selected. In the latter case, we know that the error must have occurred \ before the initially selected checkpoint. This changes the probabilities for \ failure in the various checkpoint intervals, and the computation of the \ expected cost of recovery from a given checkpoint is modified accordingly. \ The second column in the above table is adjusted as follows (if recovery from \ ", Cell[BoxData[ \(TraditionalForm\`CP\_m\)], FontSlant->"Italic"], " was unsuccessful):\n\n\t", StyleBox["P(failure after CP k ", FontSlant->"Italic"], "\[VerticalSeparator] ", StyleBox["failure before ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "m"], TraditionalForm]]], StyleBox[") =", FontSlant->"Italic"], StyleBox[" ", FontSize->14, FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(P(k) - P(m)\)\/\(1 - P(m)\)\)], FontSize->14], "\n\t\n\t", StyleBox["P(failure between ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "l"], TraditionalForm]]], StyleBox[" and ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], \(l + 1\)], TraditionalForm]]], "\[VerticalSeparator] ", StyleBox["failure before ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["CP", FontSlant->"Italic"], "m"], TraditionalForm]]], StyleBox[")", FontSlant->"Italic"], " = ", Cell[BoxData[ FormBox[ StyleBox[\(\(P(l + 1) - P(l)\)\/\(1 - P(m)\)\), FontSize->14], TraditionalForm]]], "\n\t\nAll terms have a common denominator ", StyleBox["1-P(m)", FontSlant->"Italic"], "; it can be canceled out since it does not affect the comparison of the \ expected cost of recovery from ", Cell[BoxData[ \(TraditionalForm\`CP\_k\)], FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], " and ", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{\(CP\_\(k + 1\)\), StyleBox[",", FontSlant->"Plain"], StyleBox[" ", FontSlant->"Plain"], StyleBox["respectively", FontSlant->"Plain"]}], TraditionalForm]], FontSlant->"Italic"], ". The only adjustment necessary is for the factor multiplied by ", StyleBox["T(k)", FontSlant->"Italic"], ". This results in the following revised formula for ", StyleBox["EC(k)", FontSlant->"Italic"], ":\n\t\n\t", StyleBox["EC(k) =T(k)\[CenterDot](P(1)", FontSlant->"Italic"], StyleBox["-P", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[") + ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(r = 1\)\%\(k - 1\)\), StyleBox[\(\(T(r)\)\[CenterDot]\((P(1) - P(r + 1))\)\), FontSlant->"Italic"]}], TraditionalForm]]], StyleBox[" \n\t\n", FontSlant->"Italic"], "where ", StyleBox["P", FontSlant->"Italic"], " = 0 initially, and ", StyleBox["P", FontSlant->"Italic"], " = ", StyleBox["P(m)", FontSlant->"Italic"], " if recovery from ", Cell[BoxData[ \(TraditionalForm\`CP\_m\)], FontSlant->"Italic"], " was unsuccessful." }], "Text"], Cell[TextData[{ "The function ", StyleBox["selrecovery", FontWeight->"Bold"], " computes the total cost for selective recovery once an error is detected. \ The variable ", StyleBox["total ", FontSlant->"Italic"], " denotes the number of checkpoint intervals currently established (which \ may differ from the final number in the beginning stages). The quantities ", StyleBox["probs, cost ", FontSlant->"Italic"], "(\[LongLeftRightArrow]", StyleBox["T ", FontSlant->"Italic"], "), and ", StyleBox["ec ", FontSlant->"Italic"], "are computed. Several attempts may be necessary; the variable ", StyleBox["overallsuccess", FontSlant->"Italic"], " is used to determine whether recovery has been successful or is \ impossible. After each recovery attempt, the cost for the attempt is added to \ the variable ", StyleBox["totcost", FontSlant->"Italic"], ". If another recovery attempt is necessary, then the variables ", StyleBox["P", FontSlant->"Italic"], " and ", StyleBox["ec ", FontSlant->"Italic"], "are updated." }], "Text"], Cell[BoxData[ \(\(\(selrecovery[ failuretime_]\)\(:=\)\(\n\)\(\t\)\(Module[{probs, cost, ec, ccp, overallsuccess = False, itsuccess = False, \ P\ = \ 0, totcost = 0}, \n\t\t\n\t\tprobs = Table[EDF[tslcp + \((total + 1 - k)\) cint, lagtime], {k, 1, total + 1}]; \n\t\tcost = Table[\((total + 1 - k)\) cycle + CL + tslcp, {k, 1, total + 1}]; \n\t\ \ ccp = total + 1; \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ current\ CP\ *) \n\t\n\t\tWhile[ overallsuccess\ == \ False, \n\t\t\t\ If[ccp\ == \ 1, \ itsuccess\ = \ True]; \n\t\t\t\ ec = Table[cost[\([k]\)] \((probs[\([1]\)] - P)\) + Sum[cost[\([l]\)] \((probs[\([1]\)] - probs[\([l + 1]\)])\), {l, 1, k - 1}], {k, 1, ccp}]; \n\t\t\t\n\t\t\t\ While[\((\ itsuccess\ == \ False)\)\ && \((\ ccp\ > 1)\), \ \ \ \ \ \n\t\t\t\t\ \ \ \ \ \ \(If[ ec[\([ccp]\)] >= ec[\([ccp - 1]\)], ccp\ = \ ccp - 1, itsuccess\ = \ True];\)\n\t\t\t\t\ \ \ \ \ ]; (*\ End\ While\ *) \n\t\t\t\t\t\ttotcost\ = \ totcost + cost[\([ccp]\)]; \ \ \ \ \n\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ Recovery\ attempt\ from\ CP\ ccp\ *) \n\t\t\ \ \ \ \ \ \ \ \ timeccp\ \ = \ time - \((tslcp + \((total + 1 - ccp)\) cint)\); \n\t\t\t\ \n\t\t\t\ \ \ \ \ \ If[ failuretime\ >= \ timeccp, \ \n\t\t\t\t\ \ \ \ \ \ \ \ \ \ overallsuccess\ = \ True, \ \ \ (*\ recovery\ successful\ *) \n\t\t\t\ \ \ \ \ \ (*\ else\ *) \ \ \ \ \ \ \ \ \ \ \ \n\t\t\t\t\t\ \ \ \ \ \ \ \ \ P \ = probs\ [\([ccp]\)]; \n\t\t\t\t\ \ \ \ \ \ \ \ \ \ \ ccp\ = \ ccp - 1; \n\t\t\t\t\ \ \ \ \ \ \ \ \ \ \ itsuccess\ = \ False; \n\t\t\t\t\ \ \ \ \ \ \ \ \ If[ ccp\ == \ 0, \ \ \ \ (*\ no\ recovery\ possible\ *) \n\t\t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ overallsuccess\ = \ True]\n\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ ]\ \ \ (*\ End\ If\ *) \n\t\t\t\t\ \ \ \ \ \ \ \t\n\t\t]; \ (*End\ of\ \ While\ *) \n\ttotcost\ \ \n]\)\(\ \)\( (*\ End\ of\ Module\ *) \)\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\n\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Simulation for Cost Comparison", "Subsection"], Cell[TextData[{ "To compare the two recovery methods, 30 simulations (up to time 100 hours) \ were executed for each of the values for the checkpoint interval length ", StyleBox["C. ", FontSlant->"Italic"], "The largest checkpoint interval length was chosen to be approximately the \ median of the latency times. Furthermore, the total number of checkpoint \ intervals to be maintained is computed from the latency distribution. The \ ideal situation, namely storing every single checkpoint, may not be feasible \ due to space constraints in the system. However, one would like to insure a \ certain probability of successful recovery. Using quantiles of the latency \ distribution, the program computes the total length of time that needs to be \ covered if either a 90% or 95% recovery rate is to be guaranteed and \ determines the appropriate number of checkpoint intervals. For each \ combination of interval length ", StyleBox["C", FontSlant->"Italic"], " and recovery rate, the cost for both the iterative and selective \ recovery method are written to a file for later analysis. The file names are \ constructed during the simulation to reflect the parameter values.\nThe main \ changes in the code (as compared to the file Simdata.nb) are in the part \ relating to the occurrence of an acceptance test. A new variable ", StyleBox["detect", FontSlant->"Italic"], " keeps track of which failures are detected. If one or more failures are \ detected, rollback for recovery is initiated. The critical time which \ determines whether recovery will be successful is the time of the earliest \ failure occurrence (in the case of multiple failures). The cost for the \ iterative and selective method, ", StyleBox["TCI", FontSlant->"Italic"], " and ", StyleBox["TCS", FontSlant->"Italic"], ", respectively, are updated. The modified code can be found in the file \ Simcost.nb. This file also contains the code for displaying the data to \ compare the two methods." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Results of Cost Comparison", "Subsection"], Cell[TextData[{ "The datafiles are read into ", StyleBox["Mathematica", FontSlant->"Italic"], " and displayed for each pair of parameters. Using the built-in packages \ Graphics`MultipleListPlot` and Graphics`Graphics, graphs of the data for all \ parameter combinations are displayed. 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Mabsadd L .34127 .14348 0 -2.5 Mabsadd L .34127 .14348 -1.875 0 Mabsadd L closepath F .37302 .13237 -1.875 0 Mabsadd m .37302 .13237 0 2.5 Mabsadd L .37302 .13237 1.875 0 Mabsadd L .37302 .13237 0 -2.5 Mabsadd L .37302 .13237 -1.875 0 Mabsadd L closepath F .40476 .15771 -1.875 0 Mabsadd m .40476 .15771 0 2.5 Mabsadd L .40476 .15771 1.875 0 Mabsadd L .40476 .15771 0 -2.5 Mabsadd L .40476 .15771 -1.875 0 Mabsadd L closepath F .43651 .17685 -1.875 0 Mabsadd m .43651 .17685 0 2.5 Mabsadd L .43651 .17685 1.875 0 Mabsadd L .43651 .17685 0 -2.5 Mabsadd L .43651 .17685 -1.875 0 Mabsadd L closepath F .46825 .17999 -1.875 0 Mabsadd m .46825 .17999 0 2.5 Mabsadd L .46825 .17999 1.875 0 Mabsadd L .46825 .17999 0 -2.5 Mabsadd L .46825 .17999 -1.875 0 Mabsadd L closepath F .5 .19825 -1.875 0 Mabsadd m .5 .19825 0 2.5 Mabsadd L .5 .19825 1.875 0 Mabsadd L .5 .19825 0 -2.5 Mabsadd L .5 .19825 -1.875 0 Mabsadd L closepath F .53175 .18511 -1.875 0 Mabsadd m .53175 .18511 0 2.5 Mabsadd L .53175 .18511 1.875 0 Mabsadd L .53175 .18511 0 -2.5 Mabsadd L .53175 .18511 -1.875 0 Mabsadd L closepath F .56349 .11988 -1.875 0 Mabsadd m .56349 .11988 0 2.5 Mabsadd L .56349 .11988 1.875 0 Mabsadd L .56349 .11988 0 -2.5 Mabsadd L .56349 .11988 -1.875 0 Mabsadd L closepath F .59524 .16542 -1.875 0 Mabsadd m .59524 .16542 0 2.5 Mabsadd L .59524 .16542 1.875 0 Mabsadd L .59524 .16542 0 -2.5 Mabsadd L .59524 .16542 -1.875 0 Mabsadd L closepath F .62698 .16241 -1.875 0 Mabsadd m .62698 .16241 0 2.5 Mabsadd L .62698 .16241 1.875 0 Mabsadd L .62698 .16241 0 -2.5 Mabsadd L .62698 .16241 -1.875 0 Mabsadd L closepath F .65873 .17894 -1.875 0 Mabsadd m .65873 .17894 0 2.5 Mabsadd L .65873 .17894 1.875 0 Mabsadd L .65873 .17894 0 -2.5 Mabsadd L .65873 .17894 -1.875 0 Mabsadd L closepath F .69048 .04874 -1.875 0 Mabsadd m .69048 .04874 0 2.5 Mabsadd L .69048 .04874 1.875 0 Mabsadd L .69048 .04874 0 -2.5 Mabsadd L .69048 .04874 -1.875 0 Mabsadd L closepath F .72222 .1481 -1.875 0 Mabsadd m .72222 .1481 0 2.5 Mabsadd L .72222 .1481 1.875 0 Mabsadd L .72222 .1481 0 -2.5 Mabsadd L .72222 .1481 -1.875 0 Mabsadd L closepath F .75397 .17468 -1.875 0 Mabsadd m .75397 .17468 0 2.5 Mabsadd L .75397 .17468 1.875 0 Mabsadd L .75397 .17468 0 -2.5 Mabsadd L .75397 .17468 -1.875 0 Mabsadd L closepath F .78571 .11258 -1.875 0 Mabsadd m .78571 .11258 0 2.5 Mabsadd L .78571 .11258 1.875 0 Mabsadd L .78571 .11258 0 -2.5 Mabsadd L .78571 .11258 -1.875 0 Mabsadd L closepath F .81746 .15274 -1.875 0 Mabsadd m .81746 .15274 0 2.5 Mabsadd L .81746 .15274 1.875 0 Mabsadd L .81746 .15274 0 -2.5 Mabsadd L .81746 .15274 -1.875 0 Mabsadd L closepath F .84921 .17823 -1.875 0 Mabsadd m .84921 .17823 0 2.5 Mabsadd L .84921 .17823 1.875 0 Mabsadd L .84921 .17823 0 -2.5 Mabsadd L .84921 .17823 -1.875 0 Mabsadd L closepath F .88095 .10596 -1.875 0 Mabsadd m .88095 .10596 0 2.5 Mabsadd L .88095 .10596 1.875 0 Mabsadd L .88095 .10596 0 -2.5 Mabsadd L .88095 .10596 -1.875 0 Mabsadd L closepath F .9127 .11038 -1.875 0 Mabsadd m .9127 .11038 0 2.5 Mabsadd L .9127 .11038 1.875 0 Mabsadd L .9127 .11038 0 -2.5 Mabsadd L .9127 .11038 -1.875 0 Mabsadd L closepath F .94444 .16015 -1.875 0 Mabsadd m .94444 .16015 0 2.5 Mabsadd L .94444 .16015 1.875 0 Mabsadd L .94444 .16015 0 -2.5 Mabsadd L .94444 .16015 -1.875 0 Mabsadd L closepath F .97619 .16895 -1.875 0 Mabsadd m .97619 .16895 0 2.5 Mabsadd L .97619 .16895 1.875 0 Mabsadd L .97619 .16895 0 -2.5 Mabsadd L .97619 .16895 -1.875 0 Mabsadd L closepath F .001 w [ .5 3 ] 0 Mabsdash .05556 .60332 m .0873 .26052 L .11905 .25958 L .15079 .32805 L .18254 .49557 L .21429 .60034 L .24603 .36026 L .27778 .36663 L .30952 .25231 L .34127 .31363 L .37302 .30136 L .40476 .24322 L .43651 .31471 L .46825 .3522 L .5 .46291 L .53175 .38504 L .56349 .20074 L .59524 .39009 L .62698 .26269 L .65873 .28647 L .69048 .07034 L .72222 .23178 L .75397 .30669 L .78571 .18189 L .81746 .28716 L .84921 .39458 L .88095 .18401 L .9127 .13425 L .94444 .2753 L .97619 .28254 L s .0001 w [ ] 0 setdash .05556 .60332 0 2.5 Mabsadd m .05556 .60332 -0.56518 .7779 Mabsadd L .05556 .60332 -2.37764 .77254 Mabsadd L .05556 .60332 -0.91448 -0.29713 Mabsadd L .05556 .60332 -1.46946 -2.02254 Mabsadd L .05556 .60332 0 -0.96154 Mabsadd L .05556 .60332 1.46946 -2.02254 Mabsadd L .05556 .60332 .91448 -0.29713 Mabsadd L .05556 .60332 2.37764 .77254 Mabsadd L .05556 .60332 .56518 .7779 Mabsadd L .05556 .60332 0 2.5 Mabsadd L closepath F .0873 .26052 0 2.5 Mabsadd m .0873 .26052 -0.56518 .7779 Mabsadd L .0873 .26052 -2.37764 .77254 Mabsadd L .0873 .26052 -0.91448 -0.29713 Mabsadd L .0873 .26052 -1.46946 -2.02254 Mabsadd L .0873 .26052 0 -0.96154 Mabsadd L .0873 .26052 1.46946 -2.02254 Mabsadd L .0873 .26052 .91448 -0.29713 Mabsadd L .0873 .26052 2.37764 .77254 Mabsadd L .0873 .26052 .56518 .7779 Mabsadd L .0873 .26052 0 2.5 Mabsadd L closepath F .11905 .25958 0 2.5 Mabsadd m .11905 .25958 -0.56518 .7779 Mabsadd L .11905 .25958 -2.37764 .77254 Mabsadd L .11905 .25958 -0.91448 -0.29713 Mabsadd L .11905 .25958 -1.46946 -2.02254 Mabsadd L .11905 .25958 0 -0.96154 Mabsadd L .11905 .25958 1.46946 -2.02254 Mabsadd L .11905 .25958 .91448 -0.29713 Mabsadd L .11905 .25958 2.37764 .77254 Mabsadd L .11905 .25958 .56518 .7779 Mabsadd L .11905 .25958 0 2.5 Mabsadd L closepath F .15079 .32805 0 2.5 Mabsadd m .15079 .32805 -0.56518 .7779 Mabsadd L .15079 .32805 -2.37764 .77254 Mabsadd L .15079 .32805 -0.91448 -0.29713 Mabsadd L .15079 .32805 -1.46946 -2.02254 Mabsadd L .15079 .32805 0 -0.96154 Mabsadd L .15079 .32805 1.46946 -2.02254 Mabsadd L .15079 .32805 .91448 -0.29713 Mabsadd L .15079 .32805 2.37764 .77254 Mabsadd L .15079 .32805 .56518 .7779 Mabsadd L .15079 .32805 0 2.5 Mabsadd L closepath F .18254 .49557 0 2.5 Mabsadd m .18254 .49557 -0.56518 .7779 Mabsadd L .18254 .49557 -2.37764 .77254 Mabsadd L .18254 .49557 -0.91448 -0.29713 Mabsadd L .18254 .49557 -1.46946 -2.02254 Mabsadd L .18254 .49557 0 -0.96154 Mabsadd L .18254 .49557 1.46946 -2.02254 Mabsadd L .18254 .49557 .91448 -0.29713 Mabsadd L .18254 .49557 2.37764 .77254 Mabsadd L .18254 .49557 .56518 .7779 Mabsadd L .18254 .49557 0 2.5 Mabsadd L closepath F .21429 .60034 0 2.5 Mabsadd m .21429 .60034 -0.56518 .7779 Mabsadd L .21429 .60034 -2.37764 .77254 Mabsadd L .21429 .60034 -0.91448 -0.29713 Mabsadd L .21429 .60034 -1.46946 -2.02254 Mabsadd L .21429 .60034 0 -0.96154 Mabsadd L .21429 .60034 1.46946 -2.02254 Mabsadd L .21429 .60034 .91448 -0.29713 Mabsadd L .21429 .60034 2.37764 .77254 Mabsadd L .21429 .60034 .56518 .7779 Mabsadd L .21429 .60034 0 2.5 Mabsadd L closepath F .24603 .36026 0 2.5 Mabsadd m .24603 .36026 -0.56518 .7779 Mabsadd L .24603 .36026 -2.37764 .77254 Mabsadd L .24603 .36026 -0.91448 -0.29713 Mabsadd L .24603 .36026 -1.46946 -2.02254 Mabsadd L .24603 .36026 0 -0.96154 Mabsadd L .24603 .36026 1.46946 -2.02254 Mabsadd L .24603 .36026 .91448 -0.29713 Mabsadd L .24603 .36026 2.37764 .77254 Mabsadd L .24603 .36026 .56518 .7779 Mabsadd L .24603 .36026 0 2.5 Mabsadd L closepath F .27778 .36663 0 2.5 Mabsadd m .27778 .36663 -0.56518 .7779 Mabsadd L .27778 .36663 -2.37764 .77254 Mabsadd L .27778 .36663 -0.91448 -0.29713 Mabsadd L .27778 .36663 -1.46946 -2.02254 Mabsadd L .27778 .36663 0 -0.96154 Mabsadd L .27778 .36663 1.46946 -2.02254 Mabsadd L .27778 .36663 .91448 -0.29713 Mabsadd L .27778 .36663 2.37764 .77254 Mabsadd L .27778 .36663 .56518 .7779 Mabsadd L .27778 .36663 0 2.5 Mabsadd L closepath F .30952 .25231 0 2.5 Mabsadd m .30952 .25231 -0.56518 .7779 Mabsadd L .30952 .25231 -2.37764 .77254 Mabsadd L .30952 .25231 -0.91448 -0.29713 Mabsadd L .30952 .25231 -1.46946 -2.02254 Mabsadd L .30952 .25231 0 -0.96154 Mabsadd L .30952 .25231 1.46946 -2.02254 Mabsadd L .30952 .25231 .91448 -0.29713 Mabsadd L .30952 .25231 2.37764 .77254 Mabsadd L .30952 .25231 .56518 .7779 Mabsadd L .30952 .25231 0 2.5 Mabsadd L closepath F .34127 .31363 0 2.5 Mabsadd m .34127 .31363 -0.56518 .7779 Mabsadd L .34127 .31363 -2.37764 .77254 Mabsadd L .34127 .31363 -0.91448 -0.29713 Mabsadd L .34127 .31363 -1.46946 -2.02254 Mabsadd L .34127 .31363 0 -0.96154 Mabsadd L .34127 .31363 1.46946 -2.02254 Mabsadd L .34127 .31363 .91448 -0.29713 Mabsadd L .34127 .31363 2.37764 .77254 Mabsadd L .34127 .31363 .56518 .7779 Mabsadd L .34127 .31363 0 2.5 Mabsadd L closepath F .37302 .30136 0 2.5 Mabsadd m .37302 .30136 -0.56518 .7779 Mabsadd L .37302 .30136 -2.37764 .77254 Mabsadd L .37302 .30136 -0.91448 -0.29713 Mabsadd L .37302 .30136 -1.46946 -2.02254 Mabsadd L .37302 .30136 0 -0.96154 Mabsadd L .37302 .30136 1.46946 -2.02254 Mabsadd L .37302 .30136 .91448 -0.29713 Mabsadd L .37302 .30136 2.37764 .77254 Mabsadd L .37302 .30136 .56518 .7779 Mabsadd L .37302 .30136 0 2.5 Mabsadd L closepath F .40476 .24322 0 2.5 Mabsadd m .40476 .24322 -0.56518 .7779 Mabsadd L .40476 .24322 -2.37764 .77254 Mabsadd L .40476 .24322 -0.91448 -0.29713 Mabsadd L .40476 .24322 -1.46946 -2.02254 Mabsadd L .40476 .24322 0 -0.96154 Mabsadd L .40476 .24322 1.46946 -2.02254 Mabsadd L .40476 .24322 .91448 -0.29713 Mabsadd L .40476 .24322 2.37764 .77254 Mabsadd L .40476 .24322 .56518 .7779 Mabsadd L .40476 .24322 0 2.5 Mabsadd L closepath F .43651 .31471 0 2.5 Mabsadd m .43651 .31471 -0.56518 .7779 Mabsadd L .43651 .31471 -2.37764 .77254 Mabsadd L .43651 .31471 -0.91448 -0.29713 Mabsadd L .43651 .31471 -1.46946 -2.02254 Mabsadd L .43651 .31471 0 -0.96154 Mabsadd L .43651 .31471 1.46946 -2.02254 Mabsadd L .43651 .31471 .91448 -0.29713 Mabsadd L .43651 .31471 2.37764 .77254 Mabsadd L .43651 .31471 .56518 .7779 Mabsadd L .43651 .31471 0 2.5 Mabsadd L closepath F .46825 .3522 0 2.5 Mabsadd m .46825 .3522 -0.56518 .7779 Mabsadd L .46825 .3522 -2.37764 .77254 Mabsadd L .46825 .3522 -0.91448 -0.29713 Mabsadd L .46825 .3522 -1.46946 -2.02254 Mabsadd L .46825 .3522 0 -0.96154 Mabsadd L .46825 .3522 1.46946 -2.02254 Mabsadd L .46825 .3522 .91448 -0.29713 Mabsadd L .46825 .3522 2.37764 .77254 Mabsadd L .46825 .3522 .56518 .7779 Mabsadd L .46825 .3522 0 2.5 Mabsadd L closepath F .5 .46291 0 2.5 Mabsadd m .5 .46291 -0.56518 .7779 Mabsadd L .5 .46291 -2.37764 .77254 Mabsadd L .5 .46291 -0.91448 -0.29713 Mabsadd L .5 .46291 -1.46946 -2.02254 Mabsadd L .5 .46291 0 -0.96154 Mabsadd L .5 .46291 1.46946 -2.02254 Mabsadd L .5 .46291 .91448 -0.29713 Mabsadd L .5 .46291 2.37764 .77254 Mabsadd L .5 .46291 .56518 .7779 Mabsadd L .5 .46291 0 2.5 Mabsadd L closepath F .53175 .38504 0 2.5 Mabsadd m .53175 .38504 -0.56518 .7779 Mabsadd L .53175 .38504 -2.37764 .77254 Mabsadd L .53175 .38504 -0.91448 -0.29713 Mabsadd L .53175 .38504 -1.46946 -2.02254 Mabsadd L .53175 .38504 0 -0.96154 Mabsadd L .53175 .38504 1.46946 -2.02254 Mabsadd L .53175 .38504 .91448 -0.29713 Mabsadd L .53175 .38504 2.37764 .77254 Mabsadd L .53175 .38504 .56518 .7779 Mabsadd L .53175 .38504 0 2.5 Mabsadd L closepath F .56349 .20074 0 2.5 Mabsadd m .56349 .20074 -0.56518 .7779 Mabsadd L .56349 .20074 -2.37764 .77254 Mabsadd L .56349 .20074 -0.91448 -0.29713 Mabsadd L .56349 .20074 -1.46946 -2.02254 Mabsadd L .56349 .20074 0 -0.96154 Mabsadd L .56349 .20074 1.46946 -2.02254 Mabsadd L .56349 .20074 .91448 -0.29713 Mabsadd L .56349 .20074 2.37764 .77254 Mabsadd L .56349 .20074 .56518 .7779 Mabsadd L .56349 .20074 0 2.5 Mabsadd L closepath F .59524 .39009 0 2.5 Mabsadd m .59524 .39009 -0.56518 .7779 Mabsadd L .59524 .39009 -2.37764 .77254 Mabsadd L .59524 .39009 -0.91448 -0.29713 Mabsadd L .59524 .39009 -1.46946 -2.02254 Mabsadd L .59524 .39009 0 -0.96154 Mabsadd L .59524 .39009 1.46946 -2.02254 Mabsadd L .59524 .39009 .91448 -0.29713 Mabsadd L .59524 .39009 2.37764 .77254 Mabsadd L .59524 .39009 .56518 .7779 Mabsadd L .59524 .39009 0 2.5 Mabsadd L closepath F .62698 .26269 0 2.5 Mabsadd m .62698 .26269 -0.56518 .7779 Mabsadd L .62698 .26269 -2.37764 .77254 Mabsadd L .62698 .26269 -0.91448 -0.29713 Mabsadd L .62698 .26269 -1.46946 -2.02254 Mabsadd L .62698 .26269 0 -0.96154 Mabsadd L .62698 .26269 1.46946 -2.02254 Mabsadd L .62698 .26269 .91448 -0.29713 Mabsadd L .62698 .26269 2.37764 .77254 Mabsadd L .62698 .26269 .56518 .7779 Mabsadd L .62698 .26269 0 2.5 Mabsadd L closepath F .65873 .28647 0 2.5 Mabsadd m .65873 .28647 -0.56518 .7779 Mabsadd L .65873 .28647 -2.37764 .77254 Mabsadd L .65873 .28647 -0.91448 -0.29713 Mabsadd L .65873 .28647 -1.46946 -2.02254 Mabsadd L .65873 .28647 0 -0.96154 Mabsadd L .65873 .28647 1.46946 -2.02254 Mabsadd L .65873 .28647 .91448 -0.29713 Mabsadd L .65873 .28647 2.37764 .77254 Mabsadd L .65873 .28647 .56518 .7779 Mabsadd L .65873 .28647 0 2.5 Mabsadd L closepath F .69048 .07034 0 2.5 Mabsadd m .69048 .07034 -0.56518 .7779 Mabsadd L .69048 .07034 -2.37764 .77254 Mabsadd L .69048 .07034 -0.91448 -0.29713 Mabsadd L .69048 .07034 -1.46946 -2.02254 Mabsadd L .69048 .07034 0 -0.96154 Mabsadd L .69048 .07034 1.46946 -2.02254 Mabsadd L .69048 .07034 .91448 -0.29713 Mabsadd L .69048 .07034 2.37764 .77254 Mabsadd L .69048 .07034 .56518 .7779 Mabsadd L .69048 .07034 0 2.5 Mabsadd L closepath F .72222 .23178 0 2.5 Mabsadd m .72222 .23178 -0.56518 .7779 Mabsadd L .72222 .23178 -2.37764 .77254 Mabsadd L .72222 .23178 -0.91448 -0.29713 Mabsadd L .72222 .23178 -1.46946 -2.02254 Mabsadd L .72222 .23178 0 -0.96154 Mabsadd L .72222 .23178 1.46946 -2.02254 Mabsadd L .72222 .23178 .91448 -0.29713 Mabsadd L .72222 .23178 2.37764 .77254 Mabsadd L .72222 .23178 .56518 .7779 Mabsadd L .72222 .23178 0 2.5 Mabsadd L closepath F .75397 .30669 0 2.5 Mabsadd m .75397 .30669 -0.56518 .7779 Mabsadd L .75397 .30669 -2.37764 .77254 Mabsadd L .75397 .30669 -0.91448 -0.29713 Mabsadd L .75397 .30669 -1.46946 -2.02254 Mabsadd L .75397 .30669 0 -0.96154 Mabsadd L .75397 .30669 1.46946 -2.02254 Mabsadd L .75397 .30669 .91448 -0.29713 Mabsadd L .75397 .30669 2.37764 .77254 Mabsadd L .75397 .30669 .56518 .7779 Mabsadd L .75397 .30669 0 2.5 Mabsadd L closepath F .78571 .18189 0 2.5 Mabsadd m .78571 .18189 -0.56518 .7779 Mabsadd L .78571 .18189 -2.37764 .77254 Mabsadd L .78571 .18189 -0.91448 -0.29713 Mabsadd L .78571 .18189 -1.46946 -2.02254 Mabsadd L .78571 .18189 0 -0.96154 Mabsadd L .78571 .18189 1.46946 -2.02254 Mabsadd L .78571 .18189 .91448 -0.29713 Mabsadd L .78571 .18189 2.37764 .77254 Mabsadd L .78571 .18189 .56518 .7779 Mabsadd L .78571 .18189 0 2.5 Mabsadd L closepath F .81746 .28716 0 2.5 Mabsadd m .81746 .28716 -0.56518 .7779 Mabsadd L .81746 .28716 -2.37764 .77254 Mabsadd L .81746 .28716 -0.91448 -0.29713 Mabsadd L .81746 .28716 -1.46946 -2.02254 Mabsadd L .81746 .28716 0 -0.96154 Mabsadd L .81746 .28716 1.46946 -2.02254 Mabsadd L .81746 .28716 .91448 -0.29713 Mabsadd L .81746 .28716 2.37764 .77254 Mabsadd L .81746 .28716 .56518 .7779 Mabsadd L .81746 .28716 0 2.5 Mabsadd L closepath F .84921 .39458 0 2.5 Mabsadd m .84921 .39458 -0.56518 .7779 Mabsadd L .84921 .39458 -2.37764 .77254 Mabsadd L .84921 .39458 -0.91448 -0.29713 Mabsadd L .84921 .39458 -1.46946 -2.02254 Mabsadd L .84921 .39458 0 -0.96154 Mabsadd L .84921 .39458 1.46946 -2.02254 Mabsadd L .84921 .39458 .91448 -0.29713 Mabsadd L .84921 .39458 2.37764 .77254 Mabsadd L .84921 .39458 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This is to be expected as for small checkpoint intervals \ there is a bigger chance that the failure did not occur in the most recent \ checkpoint interval and therefore the selective method has to go through \ several passes. 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The reason for this effect is that for a checkpoint interval of this length, most of the time the failure will have occurred after the last checkpoint and therefore both the selective and the iterative method roll back to he most recent checkpoint. \ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Conclusion", "Section", CellDingbat->None], Cell[TextData[{ "The above simulations indicate that for a model with independent \ exponential waiting times between events, the selective method is superior to \ the iterative method. The savings are most pronounced when the checkpoint \ interval is small. Further refinement of the model will take into account \ storage cost for the saved system states to determine the optimal length for \ the checkpoint interval ", StyleBox["C ", FontSlant->"Italic"], "to minimize the overall cost.\nTo implement the selective method, the \ latency distribution has to be determined. Since even a theoretically derived \ distribution function would require data collection to estimate parameters \ for the inter-event distributions, an approximate method can be implemented \ instead. By first using the iterative rollback mechanism with very small \ interval length ", StyleBox["C", FontSlant->"Italic"], ", the latency distribution can be estimated to within an error of ", StyleBox["C", FontSlant->"Italic"], ". ", StyleBox["A cost comparison between the selective method based on the \ approximate and the actual latency times, respectively, can be found in \ [Heubach and Pamula 1999].", FontFamily->"Times New Roman"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section", CellDingbat->None], Cell[TextData[{ "Somai, A. K. and N. H. Vaidya, ", StyleBox["Understanding Fault Tolerance and Reliability,", FontSlant->"Italic"], " Computer, 1997.\n\nWu, K. L. et al., ", StyleBox["Error Recovery in Shared Memory Multiprocessors Using Private \ Caches,", FontSlant->"Italic"], " IEEE Transactions on Parallel and Distributed Systems, April 1990.\n\n\ Chandy, K. M. et al.,", StyleBox[" Analytic Models for Rollback and Recovery Strategies in Database \ Systems,", FontSlant->"Italic"], " IEEE Transactions on Software Engineering, March 1975.\nK. Kant, ", StyleBox["A Model for Error Recovery with Global Checkpointing", FontSlant->"Italic"], ", Information Sciences, No 30, 1978.\n\nGray, J. and D. Siewiorek, ", StyleBox["High Availability Computer Systems", FontSlant->"Italic"], ", Computer, Sep. 1991.\n\nShin, K. G. and Y. H. Lee, ", StyleBox["Error Detection Process - Model, Design, and Impact on Computer \ Performance, ", FontSlant->"Italic"], " IEEE Transaction on Computers, June 1984.\n\nPamula, R. S. . ", StyleBox["Global Checkpointing for a Concurrent Processing System, ", FontSlant->"Italic"], "Int. Journal of Systems Sciences, 1990.\n\n", StyleBox["Heubach, S. and R. S. Pamula, Implementing an Approximate \ Probabilistic Algorithm for Error Recovery in Concurrent Processing Systems, \ ", FontFamily->"Times New Roman"], StyleBox["AoM/IAoM 1999 Proceedings: Computer Science", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[", Vol 17, pp. 50 - 55.", FontFamily->"Times New Roman"], StyleBox["\n", FontFamily->"Courier New", FontSize->10] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["About the Author", "Section", CellDingbat->None], Cell["\<\ Silvia Heubach is an assistant professor in the Department of Mathematics and \ Computer Science at California State University, Los Angeles. She holds a \ Ph.D. in Applied Mathematics from the University of Southern California, Los \ Angeles. Her areas of interest are probability, modeling and differential \ equations. Silvia Heubach Department of Mathematics and Computer Science California State University, Los Angeles sheubac@calstatela.edu \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Electronic Subscriptions", "Section", CellDingbat->None], Cell[TextData[{ "Included in the distribution for each electronic subscription are the \ files recovery.nb, Simviz.nb, Simdata.nb and simlagtimes.txt containing ", StyleBox["Mathematica", FontSlant->"Italic"], " code and data for the material described in this article." }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, AutoGeneratedPackage->None, WindowToolbars->"EditBar", WindowSize->{725, 668}, WindowMargins->{{Automatic, 39}, {Automatic, 21}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, Magnification->1 ] (*********************************************************************** Cached data follows. 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