gc.go

Documentation: internal/trace

     1  // Copyright 2017 The Go Authors. All rights reserved.
     2  // Use of this source code is governed by a BSD-style
     3  // license that can be found in the LICENSE file.
     4  
     5  package trace
     6  
     7  import (
     8  	"container/heap"
     9  	"math"
    10  	"sort"
    11  	"strings"
    12  	"time"
    13  )
    14  
    15  // MutatorUtil is a change in mutator utilization at a particular
    16  // time. Mutator utilization functions are represented as a
    17  // time-ordered []MutatorUtil.
    18  type MutatorUtil struct {
    19  	Time int64
    20  	// Util is the mean mutator utilization starting at Time. This
    21  	// is in the range [0, 1].
    22  	Util float64
    23  }
    24  
    25  // UtilFlags controls the behavior of MutatorUtilization.
    26  type UtilFlags int
    27  
    28  const (
    29  	// UtilSTW means utilization should account for STW events.
    30  	UtilSTW UtilFlags = 1 << iota
    31  	// UtilBackground means utilization should account for
    32  	// background mark workers.
    33  	UtilBackground
    34  	// UtilAssist means utilization should account for mark
    35  	// assists.
    36  	UtilAssist
    37  	// UtilSweep means utilization should account for sweeping.
    38  	UtilSweep
    39  
    40  	// UtilPerProc means each P should be given a separate
    41  	// utilization function. Otherwise, there is a single function
    42  	// and each P is given a fraction of the utilization.
    43  	UtilPerProc
    44  )
    45  
    46  // MutatorUtilization returns a set of mutator utilization functions
    47  // for the given trace. Each function will always end with 0
    48  // utilization. The bounds of each function are implicit in the first
    49  // and last event; outside of these bounds each function is undefined.
    50  //
    51  // If the UtilPerProc flag is not given, this always returns a single
    52  // utilization function. Otherwise, it returns one function per P.
    53  func MutatorUtilization(events []*Event, flags UtilFlags) [][]MutatorUtil {
    54  	if len(events) == 0 {
    55  		return nil
    56  	}
    57  
    58  	type perP struct {
    59  		// gc > 0 indicates that GC is active on this P.
    60  		gc int
    61  		// series the logical series number for this P. This
    62  		// is necessary because Ps may be removed and then
    63  		// re-added, and then the new P needs a new series.
    64  		series int
    65  	}
    66  	ps := []perP{}
    67  	stw := 0
    68  
    69  	out := [][]MutatorUtil{}
    70  	assists := map[uint64]bool{}
    71  	block := map[uint64]*Event{}
    72  	bgMark := map[uint64]bool{}
    73  
    74  	for _, ev := range events {
    75  		switch ev.Type {
    76  		case EvGomaxprocs:
    77  			gomaxprocs := int(ev.Args[0])
    78  			if len(ps) > gomaxprocs {
    79  				if flags&UtilPerProc != 0 {
    80  					// End each P's series.
    81  					for _, p := range ps[gomaxprocs:] {
    82  						out[p.series] = addUtil(out[p.series], MutatorUtil{ev.Ts, 0})
    83  					}
    84  				}
    85  				ps = ps[:gomaxprocs]
    86  			}
    87  			for len(ps) < gomaxprocs {
    88  				// Start new P's series.
    89  				series := 0
    90  				if flags&UtilPerProc != 0 || len(out) == 0 {
    91  					series = len(out)
    92  					out = append(out, []MutatorUtil{{ev.Ts, 1}})
    93  				}
    94  				ps = append(ps, perP{series: series})
    95  			}
    96  		case EvGCSTWStart:
    97  			if flags&UtilSTW != 0 {
    98  				stw++
    99  			}
   100  		case EvGCSTWDone:
   101  			if flags&UtilSTW != 0 {
   102  				stw--
   103  			}
   104  		case EvGCMarkAssistStart:
   105  			if flags&UtilAssist != 0 {
   106  				ps[ev.P].gc++
   107  				assists[ev.G] = true
   108  			}
   109  		case EvGCMarkAssistDone:
   110  			if flags&UtilAssist != 0 {
   111  				ps[ev.P].gc--
   112  				delete(assists, ev.G)
   113  			}
   114  		case EvGCSweepStart:
   115  			if flags&UtilSweep != 0 {
   116  				ps[ev.P].gc++
   117  			}
   118  		case EvGCSweepDone:
   119  			if flags&UtilSweep != 0 {
   120  				ps[ev.P].gc--
   121  			}
   122  		case EvGoStartLabel:
   123  			if flags&UtilBackground != 0 && strings.HasPrefix(ev.SArgs[0], "GC ") && ev.SArgs[0] != "GC (idle)" {
   124  				// Background mark worker.
   125  				//
   126  				// If we're in per-proc mode, we don't
   127  				// count dedicated workers because
   128  				// they kick all of the goroutines off
   129  				// that P, so don't directly
   130  				// contribute to goroutine latency.
   131  				if !(flags&UtilPerProc != 0 && ev.SArgs[0] == "GC (dedicated)") {
   132  					bgMark[ev.G] = true
   133  					ps[ev.P].gc++
   134  				}
   135  			}
   136  			fallthrough
   137  		case EvGoStart:
   138  			if assists[ev.G] {
   139  				// Unblocked during assist.
   140  				ps[ev.P].gc++
   141  			}
   142  			block[ev.G] = ev.Link
   143  		default:
   144  			if ev != block[ev.G] {
   145  				continue
   146  			}
   147  
   148  			if assists[ev.G] {
   149  				// Blocked during assist.
   150  				ps[ev.P].gc--
   151  			}
   152  			if bgMark[ev.G] {
   153  				// Background mark worker done.
   154  				ps[ev.P].gc--
   155  				delete(bgMark, ev.G)
   156  			}
   157  			delete(block, ev.G)
   158  		}
   159  
   160  		if flags&UtilPerProc == 0 {
   161  			// Compute the current average utilization.
   162  			if len(ps) == 0 {
   163  				continue
   164  			}
   165  			gcPs := 0
   166  			if stw > 0 {
   167  				gcPs = len(ps)
   168  			} else {
   169  				for i := range ps {
   170  					if ps[i].gc > 0 {
   171  						gcPs++
   172  					}
   173  				}
   174  			}
   175  			mu := MutatorUtil{ev.Ts, 1 - float64(gcPs)/float64(len(ps))}
   176  
   177  			// Record the utilization change. (Since
   178  			// len(ps) == len(out), we know len(out) > 0.)
   179  			out[0] = addUtil(out[0], mu)
   180  		} else {
   181  			// Check for per-P utilization changes.
   182  			for i := range ps {
   183  				p := &ps[i]
   184  				util := 1.0
   185  				if stw > 0 || p.gc > 0 {
   186  					util = 0.0
   187  				}
   188  				out[p.series] = addUtil(out[p.series], MutatorUtil{ev.Ts, util})
   189  			}
   190  		}
   191  	}
   192  
   193  	// Add final 0 utilization event to any remaining series. This
   194  	// is important to mark the end of the trace. The exact value
   195  	// shouldn't matter since no window should extend beyond this,
   196  	// but using 0 is symmetric with the start of the trace.
   197  	mu := MutatorUtil{events[len(events)-1].Ts, 0}
   198  	for i := range ps {
   199  		out[ps[i].series] = addUtil(out[ps[i].series], mu)
   200  	}
   201  	return out
   202  }
   203  
   204  func addUtil(util []MutatorUtil, mu MutatorUtil) []MutatorUtil {
   205  	if len(util) > 0 {
   206  		if mu.Util == util[len(util)-1].Util {
   207  			// No change.
   208  			return util
   209  		}
   210  		if mu.Time == util[len(util)-1].Time {
   211  			// Take the lowest utilization at a time stamp.
   212  			if mu.Util < util[len(util)-1].Util {
   213  				util[len(util)-1] = mu
   214  			}
   215  			return util
   216  		}
   217  	}
   218  	return append(util, mu)
   219  }
   220  
   221  // totalUtil is total utilization, measured in nanoseconds. This is a
   222  // separate type primarily to distinguish it from mean utilization,
   223  // which is also a float64.
   224  type totalUtil float64
   225  
   226  func totalUtilOf(meanUtil float64, dur int64) totalUtil {
   227  	return totalUtil(meanUtil * float64(dur))
   228  }
   229  
   230  // mean returns the mean utilization over dur.
   231  func (u totalUtil) mean(dur time.Duration) float64 {
   232  	return float64(u) / float64(dur)
   233  }
   234  
   235  // An MMUCurve is the minimum mutator utilization curve across
   236  // multiple window sizes.
   237  type MMUCurve struct {
   238  	series []mmuSeries
   239  }
   240  
   241  type mmuSeries struct {
   242  	util []MutatorUtil
   243  	// sums[j] is the cumulative sum of util[:j].
   244  	sums []totalUtil
   245  	// bands summarizes util in non-overlapping bands of duration
   246  	// bandDur.
   247  	bands []mmuBand
   248  	// bandDur is the duration of each band.
   249  	bandDur int64
   250  }
   251  
   252  type mmuBand struct {
   253  	// minUtil is the minimum instantaneous mutator utilization in
   254  	// this band.
   255  	minUtil float64
   256  	// cumUtil is the cumulative total mutator utilization between
   257  	// time 0 and the left edge of this band.
   258  	cumUtil totalUtil
   259  
   260  	// integrator is the integrator for the left edge of this
   261  	// band.
   262  	integrator integrator
   263  }
   264  
   265  // NewMMUCurve returns an MMU curve for the given mutator utilization
   266  // function.
   267  func NewMMUCurve(utils [][]MutatorUtil) *MMUCurve {
   268  	series := make([]mmuSeries, len(utils))
   269  	for i, util := range utils {
   270  		series[i] = newMMUSeries(util)
   271  	}
   272  	return &MMUCurve{series}
   273  }
   274  
   275  // bandsPerSeries is the number of bands to divide each series into.
   276  // This is only changed by tests.
   277  var bandsPerSeries = 1000
   278  
   279  func newMMUSeries(util []MutatorUtil) mmuSeries {
   280  	// Compute cumulative sum.
   281  	sums := make([]totalUtil, len(util))
   282  	var prev MutatorUtil
   283  	var sum totalUtil
   284  	for j, u := range util {
   285  		sum += totalUtilOf(prev.Util, u.Time-prev.Time)
   286  		sums[j] = sum
   287  		prev = u
   288  	}
   289  
   290  	// Divide the utilization curve up into equal size
   291  	// non-overlapping "bands" and compute a summary for each of
   292  	// these bands.
   293  	//
   294  	// Compute the duration of each band.
   295  	numBands := bandsPerSeries
   296  	if numBands > len(util) {
   297  		// There's no point in having lots of bands if there
   298  		// aren't many events.
   299  		numBands = len(util)
   300  	}
   301  	dur := util[len(util)-1].Time - util[0].Time
   302  	bandDur := (dur + int64(numBands) - 1) / int64(numBands)
   303  	if bandDur < 1 {
   304  		bandDur = 1
   305  	}
   306  	// Compute the bands. There are numBands+1 bands in order to
   307  	// record the final cumulative sum.
   308  	bands := make([]mmuBand, numBands+1)
   309  	s := mmuSeries{util, sums, bands, bandDur}
   310  	leftSum := integrator{&s, 0}
   311  	for i := range bands {
   312  		startTime, endTime := s.bandTime(i)
   313  		cumUtil := leftSum.advance(startTime)
   314  		predIdx := leftSum.pos
   315  		minUtil := 1.0
   316  		for i := predIdx; i < len(util) && util[i].Time < endTime; i++ {
   317  			minUtil = math.Min(minUtil, util[i].Util)
   318  		}
   319  		bands[i] = mmuBand{minUtil, cumUtil, leftSum}
   320  	}
   321  
   322  	return s
   323  }
   324  
   325  func (s *mmuSeries) bandTime(i int) (start, end int64) {
   326  	start = int64(i)*s.bandDur + s.util[0].Time
   327  	end = start + s.bandDur
   328  	return
   329  }
   330  
   331  type bandUtil struct {
   332  	// Utilization series index
   333  	series int
   334  	// Band index
   335  	i int
   336  	// Lower bound of mutator utilization for all windows
   337  	// with a left edge in this band.
   338  	utilBound float64
   339  }
   340  
   341  type bandUtilHeap []bandUtil
   342  
   343  func (h bandUtilHeap) Len() int {
   344  	return len(h)
   345  }
   346  
   347  func (h bandUtilHeap) Less(i, j int) bool {
   348  	return h[i].utilBound < h[j].utilBound
   349  }
   350  
   351  func (h bandUtilHeap) Swap(i, j int) {
   352  	h[i], h[j] = h[j], h[i]
   353  }
   354  
   355  func (h *bandUtilHeap) Push(x interface{}) {
   356  	*h = append(*h, x.(bandUtil))
   357  }
   358  
   359  func (h *bandUtilHeap) Pop() interface{} {
   360  	x := (*h)[len(*h)-1]
   361  	*h = (*h)[:len(*h)-1]
   362  	return x
   363  }
   364  
   365  // UtilWindow is a specific window at Time.
   366  type UtilWindow struct {
   367  	Time int64
   368  	// MutatorUtil is the mean mutator utilization in this window.
   369  	MutatorUtil float64
   370  }
   371  
   372  type utilHeap []UtilWindow
   373  
   374  func (h utilHeap) Len() int {
   375  	return len(h)
   376  }
   377  
   378  func (h utilHeap) Less(i, j int) bool {
   379  	if h[i].MutatorUtil != h[j].MutatorUtil {
   380  		return h[i].MutatorUtil > h[j].MutatorUtil
   381  	}
   382  	return h[i].Time > h[j].Time
   383  }
   384  
   385  func (h utilHeap) Swap(i, j int) {
   386  	h[i], h[j] = h[j], h[i]
   387  }
   388  
   389  func (h *utilHeap) Push(x interface{}) {
   390  	*h = append(*h, x.(UtilWindow))
   391  }
   392  
   393  func (h *utilHeap) Pop() interface{} {
   394  	x := (*h)[len(*h)-1]
   395  	*h = (*h)[:len(*h)-1]
   396  	return x
   397  }
   398  
   399  // An accumulator takes a windowed mutator utilization function and
   400  // tracks various statistics for that function.
   401  type accumulator struct {
   402  	mmu float64
   403  
   404  	// bound is the mutator utilization bound where adding any
   405  	// mutator utilization above this bound cannot affect the
   406  	// accumulated statistics.
   407  	bound float64
   408  
   409  	// Worst N window tracking
   410  	nWorst int
   411  	wHeap  utilHeap
   412  
   413  	// Mutator utilization distribution tracking
   414  	mud *mud
   415  	// preciseMass is the distribution mass that must be precise
   416  	// before accumulation is stopped.
   417  	preciseMass float64
   418  	// lastTime and lastMU are the previous point added to the
   419  	// windowed mutator utilization function.
   420  	lastTime int64
   421  	lastMU   float64
   422  }
   423  
   424  // resetTime declares a discontinuity in the windowed mutator
   425  // utilization function by resetting the current time.
   426  func (acc *accumulator) resetTime() {
   427  	// This only matters for distribution collection, since that's
   428  	// the only thing that depends on the progression of the
   429  	// windowed mutator utilization function.
   430  	acc.lastTime = math.MaxInt64
   431  }
   432  
   433  // addMU adds a point to the windowed mutator utilization function at
   434  // (time, mu). This must be called for monotonically increasing values
   435  // of time.
   436  //
   437  // It returns true if further calls to addMU would be pointless.
   438  func (acc *accumulator) addMU(time int64, mu float64, window time.Duration) bool {
   439  	if mu < acc.mmu {
   440  		acc.mmu = mu
   441  	}
   442  	acc.bound = acc.mmu
   443  
   444  	if acc.nWorst == 0 {
   445  		// If the minimum has reached zero, it can't go any
   446  		// lower, so we can stop early.
   447  		return mu == 0
   448  	}
   449  
   450  	// Consider adding this window to the n worst.
   451  	if len(acc.wHeap) < acc.nWorst || mu < acc.wHeap[0].MutatorUtil {
   452  		// This window is lower than the K'th worst window.
   453  		//
   454  		// Check if there's any overlapping window
   455  		// already in the heap and keep whichever is
   456  		// worse.
   457  		for i, ui := range acc.wHeap {
   458  			if time+int64(window) > ui.Time && ui.Time+int64(window) > time {
   459  				if ui.MutatorUtil <= mu {
   460  					// Keep the first window.
   461  					goto keep
   462  				} else {
   463  					// Replace it with this window.
   464  					heap.Remove(&acc.wHeap, i)
   465  					break
   466  				}
   467  			}
   468  		}
   469  
   470  		heap.Push(&acc.wHeap, UtilWindow{time, mu})
   471  		if len(acc.wHeap) > acc.nWorst {
   472  			heap.Pop(&acc.wHeap)
   473  		}
   474  	keep:
   475  	}
   476  
   477  	if len(acc.wHeap) < acc.nWorst {
   478  		// We don't have N windows yet, so keep accumulating.
   479  		acc.bound = 1.0
   480  	} else {
   481  		// Anything above the least worst window has no effect.
   482  		acc.bound = math.Max(acc.bound, acc.wHeap[0].MutatorUtil)
   483  	}
   484  
   485  	if acc.mud != nil {
   486  		if acc.lastTime != math.MaxInt64 {
   487  			// Update distribution.
   488  			acc.mud.add(acc.lastMU, mu, float64(time-acc.lastTime))
   489  		}
   490  		acc.lastTime, acc.lastMU = time, mu
   491  		if _, mudBound, ok := acc.mud.approxInvCumulativeSum(); ok {
   492  			acc.bound = math.Max(acc.bound, mudBound)
   493  		} else {
   494  			// We haven't accumulated enough total precise
   495  			// mass yet to even reach our goal, so keep
   496  			// accumulating.
   497  			acc.bound = 1
   498  		}
   499  		// It's not worth checking percentiles every time, so
   500  		// just keep accumulating this band.
   501  		return false
   502  	}
   503  
   504  	// If we've found enough 0 utilizations, we can stop immediately.
   505  	return len(acc.wHeap) == acc.nWorst && acc.wHeap[0].MutatorUtil == 0
   506  }
   507  
   508  // MMU returns the minimum mutator utilization for the given time
   509  // window. This is the minimum utilization for all windows of this
   510  // duration across the execution. The returned value is in the range
   511  // [0, 1].
   512  func (c *MMUCurve) MMU(window time.Duration) (mmu float64) {
   513  	acc := accumulator{mmu: 1.0, bound: 1.0}
   514  	c.mmu(window, &acc)
   515  	return acc.mmu
   516  }
   517  
   518  // Examples returns n specific examples of the lowest mutator
   519  // utilization for the given window size. The returned windows will be
   520  // disjoint (otherwise there would be a huge number of
   521  // mostly-overlapping windows at the single lowest point). There are
   522  // no guarantees on which set of disjoint windows this returns.
   523  func (c *MMUCurve) Examples(window time.Duration, n int) (worst []UtilWindow) {
   524  	acc := accumulator{mmu: 1.0, bound: 1.0, nWorst: n}
   525  	c.mmu(window, &acc)
   526  	sort.Sort(sort.Reverse(acc.wHeap))
   527  	return ([]UtilWindow)(acc.wHeap)
   528  }
   529  
   530  // MUD returns mutator utilization distribution quantiles for the
   531  // given window size.
   532  //
   533  // The mutator utilization distribution is the distribution of mean
   534  // mutator utilization across all windows of the given window size in
   535  // the trace.
   536  //
   537  // The minimum mutator utilization is the minimum (0th percentile) of
   538  // this distribution. (However, if only the minimum is desired, it's
   539  // more efficient to use the MMU method.)
   540  func (c *MMUCurve) MUD(window time.Duration, quantiles []float64) []float64 {
   541  	if len(quantiles) == 0 {
   542  		return []float64{}
   543  	}
   544  
   545  	// Each unrefined band contributes a known total mass to the
   546  	// distribution (bandDur except at the end), but in an unknown
   547  	// way. However, we know that all the mass it contributes must
   548  	// be at or above its worst-case mean mutator utilization.
   549  	//
   550  	// Hence, we refine bands until the highest desired
   551  	// distribution quantile is less than the next worst-case mean
   552  	// mutator utilization. At this point, all further
   553  	// contributions to the distribution must be beyond the
   554  	// desired quantile and hence cannot affect it.
   555  	//
   556  	// First, find the highest desired distribution quantile.
   557  	maxQ := quantiles[0]
   558  	for _, q := range quantiles {
   559  		if q > maxQ {
   560  			maxQ = q
   561  		}
   562  	}
   563  	// The distribution's mass is in units of time (it's not
   564  	// normalized because this would make it more annoying to
   565  	// account for future contributions of unrefined bands). The
   566  	// total final mass will be the duration of the trace itself
   567  	// minus the window size. Using this, we can compute the mass
   568  	// corresponding to quantile maxQ.
   569  	var duration int64
   570  	for _, s := range c.series {
   571  		duration1 := s.util[len(s.util)-1].Time - s.util[0].Time
   572  		if duration1 >= int64(window) {
   573  			duration += duration1 - int64(window)
   574  		}
   575  	}
   576  	qMass := float64(duration) * maxQ
   577  
   578  	// Accumulate the MUD until we have precise information for
   579  	// everything to the left of qMass.
   580  	acc := accumulator{mmu: 1.0, bound: 1.0, preciseMass: qMass, mud: new(mud)}
   581  	acc.mud.setTrackMass(qMass)
   582  	c.mmu(window, &acc)
   583  
   584  	// Evaluate the quantiles on the accumulated MUD.
   585  	out := make([]float64, len(quantiles))
   586  	for i := range out {
   587  		mu, _ := acc.mud.invCumulativeSum(float64(duration) * quantiles[i])
   588  		if math.IsNaN(mu) {
   589  			// There are a few legitimate ways this can
   590  			// happen:
   591  			//
   592  			// 1. If the window is the full trace
   593  			// duration, then the windowed MU function is
   594  			// only defined at a single point, so the MU
   595  			// distribution is not well-defined.
   596  			//
   597  			// 2. If there are no events, then the MU
   598  			// distribution has no mass.
   599  			//
   600  			// Either way, all of the quantiles will have
   601  			// converged toward the MMU at this point.
   602  			mu = acc.mmu
   603  		}
   604  		out[i] = mu
   605  	}
   606  	return out
   607  }
   608  
   609  func (c *MMUCurve) mmu(window time.Duration, acc *accumulator) {
   610  	if window <= 0 {
   611  		acc.mmu = 0
   612  		return
   613  	}
   614  
   615  	var bandU bandUtilHeap
   616  	windows := make([]time.Duration, len(c.series))
   617  	for i, s := range c.series {
   618  		windows[i] = window
   619  		if max := time.Duration(s.util[len(s.util)-1].Time - s.util[0].Time); window > max {
   620  			windows[i] = max
   621  		}
   622  
   623  		bandU1 := bandUtilHeap(s.mkBandUtil(i, windows[i]))
   624  		if bandU == nil {
   625  			bandU = bandU1
   626  		} else {
   627  			bandU = append(bandU, bandU1...)
   628  		}
   629  	}
   630  
   631  	// Process bands from lowest utilization bound to highest.
   632  	heap.Init(&bandU)
   633  
   634  	// Refine each band into a precise window and MMU until
   635  	// refining the next lowest band can no longer affect the MMU
   636  	// or windows.
   637  	for len(bandU) > 0 && bandU[0].utilBound < acc.bound {
   638  		i := bandU[0].series
   639  		c.series[i].bandMMU(bandU[0].i, windows[i], acc)
   640  		heap.Pop(&bandU)
   641  	}
   642  }
   643  
   644  func (c *mmuSeries) mkBandUtil(series int, window time.Duration) []bandUtil {
   645  	// For each band, compute the worst-possible total mutator
   646  	// utilization for all windows that start in that band.
   647  
   648  	// minBands is the minimum number of bands a window can span
   649  	// and maxBands is the maximum number of bands a window can
   650  	// span in any alignment.
   651  	minBands := int((int64(window) + c.bandDur - 1) / c.bandDur)
   652  	maxBands := int((int64(window) + 2*(c.bandDur-1)) / c.bandDur)
   653  	if window > 1 && maxBands < 2 {
   654  		panic("maxBands < 2")
   655  	}
   656  	tailDur := int64(window) % c.bandDur
   657  	nUtil := len(c.bands) - maxBands + 1
   658  	if nUtil < 0 {
   659  		nUtil = 0
   660  	}
   661  	bandU := make([]bandUtil, nUtil)
   662  	for i := range bandU {
   663  		// To compute the worst-case MU, we assume the minimum
   664  		// for any bands that are only partially overlapped by
   665  		// some window and the mean for any bands that are
   666  		// completely covered by all windows.
   667  		var util totalUtil
   668  
   669  		// Find the lowest and second lowest of the partial
   670  		// bands.
   671  		l := c.bands[i].minUtil
   672  		r1 := c.bands[i+minBands-1].minUtil
   673  		r2 := c.bands[i+maxBands-1].minUtil
   674  		minBand := math.Min(l, math.Min(r1, r2))
   675  		// Assume the worst window maximally overlaps the
   676  		// worst minimum and then the rest overlaps the second
   677  		// worst minimum.
   678  		if minBands == 1 {
   679  			util += totalUtilOf(minBand, int64(window))
   680  		} else {
   681  			util += totalUtilOf(minBand, c.bandDur)
   682  			midBand := 0.0
   683  			switch {
   684  			case minBand == l:
   685  				midBand = math.Min(r1, r2)
   686  			case minBand == r1:
   687  				midBand = math.Min(l, r2)
   688  			case minBand == r2:
   689  				midBand = math.Min(l, r1)
   690  			}
   691  			util += totalUtilOf(midBand, tailDur)
   692  		}
   693  
   694  		// Add the total mean MU of bands that are completely
   695  		// overlapped by all windows.
   696  		if minBands > 2 {
   697  			util += c.bands[i+minBands-1].cumUtil - c.bands[i+1].cumUtil
   698  		}
   699  
   700  		bandU[i] = bandUtil{series, i, util.mean(window)}
   701  	}
   702  
   703  	return bandU
   704  }
   705  
   706  // bandMMU computes the precise minimum mutator utilization for
   707  // windows with a left edge in band bandIdx.
   708  func (c *mmuSeries) bandMMU(bandIdx int, window time.Duration, acc *accumulator) {
   709  	util := c.util
   710  
   711  	// We think of the mutator utilization over time as the
   712  	// box-filtered utilization function, which we call the
   713  	// "windowed mutator utilization function". The resulting
   714  	// function is continuous and piecewise linear (unless
   715  	// window==0, which we handle elsewhere), where the boundaries
   716  	// between segments occur when either edge of the window
   717  	// encounters a change in the instantaneous mutator
   718  	// utilization function. Hence, the minimum of this function
   719  	// will always occur when one of the edges of the window
   720  	// aligns with a utilization change, so these are the only
   721  	// points we need to consider.
   722  	//
   723  	// We compute the mutator utilization function incrementally
   724  	// by tracking the integral from t=0 to the left edge of the
   725  	// window and to the right edge of the window.
   726  	left := c.bands[bandIdx].integrator
   727  	right := left
   728  	time, endTime := c.bandTime(bandIdx)
   729  	if utilEnd := util[len(util)-1].Time - int64(window); utilEnd < endTime {
   730  		endTime = utilEnd
   731  	}
   732  	acc.resetTime()
   733  	for {
   734  		// Advance edges to time and time+window.
   735  		mu := (right.advance(time+int64(window)) - left.advance(time)).mean(window)
   736  		if acc.addMU(time, mu, window) {
   737  			break
   738  		}
   739  		if time == endTime {
   740  			break
   741  		}
   742  
   743  		// The maximum slope of the windowed mutator
   744  		// utilization function is 1/window, so we can always
   745  		// advance the time by at least (mu - mmu) * window
   746  		// without dropping below mmu.
   747  		minTime := time + int64((mu-acc.bound)*float64(window))
   748  
   749  		// Advance the window to the next time where either
   750  		// the left or right edge of the window encounters a
   751  		// change in the utilization curve.
   752  		if t1, t2 := left.next(time), right.next(time+int64(window))-int64(window); t1 < t2 {
   753  			time = t1
   754  		} else {
   755  			time = t2
   756  		}
   757  		if time < minTime {
   758  			time = minTime
   759  		}
   760  		if time >= endTime {
   761  			// For MMUs we could stop here, but for MUDs
   762  			// it's important that we span the entire
   763  			// band.
   764  			time = endTime
   765  		}
   766  	}
   767  }
   768  
   769  // An integrator tracks a position in a utilization function and
   770  // integrates it.
   771  type integrator struct {
   772  	u *mmuSeries
   773  	// pos is the index in u.util of the current time's non-strict
   774  	// predecessor.
   775  	pos int
   776  }
   777  
   778  // advance returns the integral of the utilization function from 0 to
   779  // time. advance must be called on monotonically increasing values of
   780  // times.
   781  func (in *integrator) advance(time int64) totalUtil {
   782  	util, pos := in.u.util, in.pos
   783  	// Advance pos until pos+1 is time's strict successor (making
   784  	// pos time's non-strict predecessor).
   785  	//
   786  	// Very often, this will be nearby, so we optimize that case,
   787  	// but it may be arbitrarily far away, so we handled that
   788  	// efficiently, too.
   789  	const maxSeq = 8
   790  	if pos+maxSeq < len(util) && util[pos+maxSeq].Time > time {
   791  		// Nearby. Use a linear scan.
   792  		for pos+1 < len(util) && util[pos+1].Time <= time {
   793  			pos++
   794  		}
   795  	} else {
   796  		// Far. Binary search for time's strict successor.
   797  		l, r := pos, len(util)
   798  		for l < r {
   799  			h := int(uint(l+r) >> 1)
   800  			if util[h].Time <= time {
   801  				l = h + 1
   802  			} else {
   803  				r = h
   804  			}
   805  		}
   806  		pos = l - 1 // Non-strict predecessor.
   807  	}
   808  	in.pos = pos
   809  	var partial totalUtil
   810  	if time != util[pos].Time {
   811  		partial = totalUtilOf(util[pos].Util, time-util[pos].Time)
   812  	}
   813  	return in.u.sums[pos] + partial
   814  }
   815  
   816  // next returns the smallest time t' > time of a change in the
   817  // utilization function.
   818  func (in *integrator) next(time int64) int64 {
   819  	for _, u := range in.u.util[in.pos:] {
   820  		if u.Time > time {
   821  			return u.Time
   822  		}
   823  	}
   824  	return 1<<63 - 1
   825  }
   826
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