diff --git a/cmd/gaia/app/sim_test.go b/cmd/gaia/app/sim_test.go
index 5872b1aa621c..f8ebb62c0a71 100644
--- a/cmd/gaia/app/sim_test.go
+++ b/cmd/gaia/app/sim_test.go
@@ -120,7 +120,7 @@ func appStateRandomizedFn(r *rand.Rand, accs []simulation.Account, genesisTimest
 		if int64(i) > numInitiallyBonded && r.Intn(100) < 50 {
 			var (
 				vacc    auth.VestingAccount
-				endTime int
+				endTime int64
 			)
 
 			startTime := genesisTimestamp.Unix()
@@ -128,15 +128,19 @@ func appStateRandomizedFn(r *rand.Rand, accs []simulation.Account, genesisTimest
 			// Allow for some vesting accounts to vest very quickly while others very
 			// slowly.
 			if r.Intn(100) < 50 {
-				endTime = randIntBetween(r, int(startTime), int(startTime+(60*60*24*30)))
+				endTime = int64(simulation.RandIntBetween(r, int(startTime), int(startTime+(60*60*24*30))))
 			} else {
-				endTime = randIntBetween(r, int(startTime), int(startTime+(60*60*12)))
+				endTime = int64(simulation.RandIntBetween(r, int(startTime), int(startTime+(60*60*12))))
+			}
+
+			if startTime == endTime {
+				endTime += 1
 			}
 
 			if r.Intn(100) < 50 {
-				vacc = auth.NewContinuousVestingAccount(&bacc, startTime, int64(endTime))
+				vacc = auth.NewContinuousVestingAccount(&bacc, startTime, endTime)
 			} else {
-				vacc = auth.NewDelayedVestingAccount(&bacc, int64(endTime))
+				vacc = auth.NewDelayedVestingAccount(&bacc, endTime)
 			}
 
 			gacc = NewGenesisAccountI(vacc)
@@ -149,11 +153,11 @@ func appStateRandomizedFn(r *rand.Rand, accs []simulation.Account, genesisTimest
 
 	authGenesis := auth.GenesisState{
 		Params: auth.Params{
-			MaxMemoCharacters:      uint64(randIntBetween(r, 100, 200)),
+			MaxMemoCharacters:      uint64(simulation.RandIntBetween(r, 100, 200)),
 			TxSigLimit:             uint64(r.Intn(7) + 1),
-			TxSizeCostPerByte:      uint64(randIntBetween(r, 5, 15)),
-			SigVerifyCostED25519:   uint64(randIntBetween(r, 500, 1000)),
-			SigVerifyCostSecp256k1: uint64(randIntBetween(r, 500, 1000)),
+			TxSizeCostPerByte:      uint64(simulation.RandIntBetween(r, 5, 15)),
+			SigVerifyCostED25519:   uint64(simulation.RandIntBetween(r, 500, 1000)),
+			SigVerifyCostSecp256k1: uint64(simulation.RandIntBetween(r, 500, 1000)),
 		},
 	}
 	fmt.Printf("Selected randomly generated auth parameters:\n\t%+v\n", authGenesis)
@@ -183,7 +187,7 @@ func appStateRandomizedFn(r *rand.Rand, accs []simulation.Account, genesisTimest
 	stakingGenesis := staking.GenesisState{
 		Pool: staking.InitialPool(),
 		Params: staking.Params{
-			UnbondingTime: time.Duration(randIntBetween(r, 60, 60*60*24*3*2)) * time.Second,
+			UnbondingTime: time.Duration(simulation.RandIntBetween(r, 60, 60*60*24*3*2)) * time.Second,
 			MaxValidators: uint16(r.Intn(250) + 1),
 			BondDenom:     sdk.DefaultBondDenom,
 		},
@@ -193,9 +197,9 @@ func appStateRandomizedFn(r *rand.Rand, accs []simulation.Account, genesisTimest
 	slashingGenesis := slashing.GenesisState{
 		Params: slashing.Params{
 			MaxEvidenceAge:          stakingGenesis.Params.UnbondingTime,
-			SignedBlocksWindow:      int64(randIntBetween(r, 10, 1000)),
+			SignedBlocksWindow:      int64(simulation.RandIntBetween(r, 10, 1000)),
 			MinSignedPerWindow:      sdk.NewDecWithPrec(int64(r.Intn(10)), 1),
-			DowntimeJailDuration:    time.Duration(randIntBetween(r, 60, 60*60*24)) * time.Second,
+			DowntimeJailDuration:    time.Duration(simulation.RandIntBetween(r, 60, 60*60*24)) * time.Second,
 			SlashFractionDoubleSign: sdk.NewDec(1).Quo(sdk.NewDec(int64(r.Intn(50) + 1))),
 			SlashFractionDowntime:   sdk.NewDec(1).Quo(sdk.NewDec(int64(r.Intn(200) + 1))),
 		},
@@ -270,10 +274,6 @@ func appStateFn(r *rand.Rand, accs []simulation.Account, genesisTimestamp time.T
 	return appStateRandomizedFn(r, accs, genesisTimestamp)
 }
 
-func randIntBetween(r *rand.Rand, min, max int) int {
-	return r.Intn(max-min) + min
-}
-
 func testAndRunTxs(app *GaiaApp) []simulation.WeightedOperation {
 	return []simulation.WeightedOperation{
 		{5, authsim.SimulateDeductFee(app.accountKeeper, app.feeCollectionKeeper)},
diff --git a/x/simulation/params.go b/x/simulation/params.go
index 8499e6c1189b..6202ec6fef37 100644
--- a/x/simulation/params.go
+++ b/x/simulation/params.go
@@ -60,7 +60,7 @@ func DefaultParams() Params {
 func RandomParams(r *rand.Rand) Params {
 	return Params{
 		PastEvidenceFraction:      r.Float64(),
-		NumKeys:                   r.Intn(250),
+		NumKeys:                   RandIntBetween(r, 2, 250),
 		EvidenceFraction:          r.Float64(),
 		InitialLivenessWeightings: []int{r.Intn(80), r.Intn(10), r.Intn(10)},
 		LivenessTransitionMatrix:  defaultLivenessTransitionMatrix,
diff --git a/x/simulation/rand_util.go b/x/simulation/rand_util.go
index a9b32ca48dd4..9677f2a612e6 100644
--- a/x/simulation/rand_util.go
+++ b/x/simulation/rand_util.go
@@ -72,6 +72,11 @@ func RandTimestamp(r *rand.Rand) time.Time {
 	return time.Unix(unixTime, 0)
 }
 
+// RandIntBetween returns a random int between two numbers inclusively.
+func RandIntBetween(r *rand.Rand, min, max int) int {
+	return r.Intn(max-min) + min
+}
+
 // Derive a new rand deterministically from a rand.
 // Unlike rand.New(rand.NewSource(seed)), the result is "more random"
 // depending on the source and state of r.