You can use TemporalData to construct your input data and then use TimeSeriesRescale on individual time series to get a perspective look/feel:
numberofseries = 20;
tds = Table[TemporalData[Table[Sin[x + i] + 2*i, {x, 0, 10, 0.1}], {0, 10}],
{i, numberofseries}];
ClearAll[xaxis]
xaxis[a_, b_] := {GrayLevel[.4], Thickness -> Absolute[0.2],
{Line[#[[{1, -1}]]], Line[{#, Offset[{0, -5}, #]}] & /@ #,
Text[#[[1]], Offset[{0, -15}, #]] & /@ #} & @ Thread[{Range[a, b], 0}]};
ListLinePlot[tds, ImageSize -> Large, Axes -> False,
Epilog -> xaxis[0, 10], AspectRatio -> 1/2, ImagePadding -> 20,
PlotRangeClipping -> False, Ticks -> {Range[0, 10], All}]
ClearAll[rescaleTD]
rescaleTD[ls_: .1, rs_: .1] := MapIndexed[TimeSeriesRescale[#,
Flatten[# /@ {"FirstTimes", "LastTimes"}] + {ls, rs} #2[[1]]] &, #] &;
Grid @ (ListLinePlot[#, ImageSize -> Medium, Axes -> False,
Epilog -> xaxis, AspectRatio -> 1/2, ImagePadding -> 20,
PlotRangeClipping -> False,
Ticks -> {Range[0, 10], All}] & /@
{ rescaleTD[#, #]@tds, rescaleTD[#, -#]@tds, rescaleTD[#, -#/2]@tds} & /@ {.1, .15, .2})
