Started work on exercise 8b.

code/8b-report.R

@@ -0,0 +1,3 @@
+mm1_var <- var(sapply(1:100, \(...) calc_stats_mm1(...)$avg_delay))
+mg1_var1 <- var(sapply(1:100, \(...) calc_stats_mg1(...)$avg_delay))
+mg1_var2 <- var(sapply(1:100, \(...) calc_stats_mg1(...)$avg_delay))

code/8b.R

@@ -1,13 +1,16 @@
 source("code/8.R")
 
-set.seed(3)
+set.seed(0)
 
 stopping_condition <- 10000
 mm1_var <- var(sapply(1:100, function(...) calc_stats_mm1(0.5, 1, 0.99, 101, 0.01, stopping_condition)$avg_delay))
-print(mm1_var)
-
 mg1_var1 <- var(sapply(1:100, function(...) calc_stats_mg1(0.5, 1, 0.9, 11, 0.1, stopping_condition)$avg_delay))
-print(mg1_var1)
-
 mg1_var2 <- var(sapply(1:100, function(...) calc_stats_mg1(0.5, 1, 0.99, 101, 0.01, stopping_condition)$avg_delay))
-print(mg1_var2)
+
+output <- list(
+  mm1_var = signif(mm1_var, 4),
+  mg1_1_var = signif(mg1_var1, 4),
+  mg1_2_var = signif(mg1_var2, 4)
+)
+str(output)
+write.table(output, "output/8b.csv", sep = "\t", row.names = FALSE)

typst/exercises/8.typ

@@ -3,7 +3,7 @@
 
 ==== a.
 
-#let results = csv("/output/8a.csv", delimiter: "\t")
+#let results_a = csv("/output/8a.csv", delimiter: "\t")
 
 #indent_par[The following table 15 contains the results of the script. The simulated results have been run 10 times, and are displayed as a confidence interval of 95%.]
 
@@ -29,20 +29,20 @@
 		[ `M/M/1` ],
 		rowspanx(3)[ 0.5 ],
 		colspanx(4)[ N/A ],
-		[ #results.at(1).at(0) .. #results.at(1).at(1) ],
-		[ #results.at(1).at(2) ],
-		[ #results.at(1).at(3) ],
+		[ #results_a.at(1).at(0) .. #results_a.at(1).at(1) ],
+		[ #results_a.at(1).at(2) ],
+		[ #results_a.at(1).at(3) ],
 
 		rowspanx(2)[ `M/G/1` ],
 		[ 1 ], [ 0.9 ], [ 11 ], [ 0.1 ],
-		[ #results.at(1).at(4) .. #results.at(1).at(5) ],
-		[ #results.at(1).at(6) ],
-		[ #results.at(1).at(7) ],
+		[ #results_a.at(1).at(4) .. #results_a.at(1).at(5) ],
+		[ #results_a.at(1).at(6) ],
+		[ #results_a.at(1).at(7) ],
 
 		[ 1 ], [ 0.99 ], [ 101 ], [ 0.01 ],
-		[ #results.at(1).at(8) .. #results.at(1).at(9) ],
-		[ #results.at(1).at(10) ],
-		[ #results.at(1).at(11) ],
+		[ #results_a.at(1).at(8) .. #results_a.at(1).at(9) ],
+		[ #results_a.at(1).at(10) ],
+		[ #results_a.at(1).at(11) ],
 	)),
 	kind: table,
 	caption: [Results]
@@ -51,3 +51,55 @@
 #indent_par[The simulated average delay results line up quite well with the calculated theoretical ones, with all values inside of the confidence interval.]
 
 #indent_par[We can also conclude that the higher the workload variability ($C^2$), the higher the average delay experienced by the system.]
+
+==== b.
+
+#let results_b = csv("/output/8b.csv", delimiter: "\t")
+
+#indent_par[The following code 8 contains out approach to calculate the variance. The functions `calc_stats_mm1` and `calc_stats_mg1` simulate the corresponding systems and return a list where `$avg_delay` contains the simulated average delay.]
+
+#code_figure(
+  text(size: 0.8em, raw(read("/code/8b-report.R"), lang: "R", block: true)),
+  caption: "Code for calculating the variance of the systems",
+)
+
+#indent_par[The following table 16 contains the results to our approach:]
+
+#figure(
+	pad(1em, tablex(
+		columns: (auto, auto, auto, auto, auto, auto, auto, auto),
+		align: center + horizon,
+
+		[ System ],
+		[ $ρ$ ],
+		[ $S_1$ ],
+		[ $p_1$ ],
+		[ $S_2$ ],
+		[ $p_2$ ],
+		[ Variance ],
+		[ $C^2$ ],
+
+		[ `M/M/1` ],
+		rowspanx(3)[ 0.5 ],
+		colspanx(4)[ N/A ],
+		[ #results_b.at(1).at(0) ],
+		[ #results_a.at(1).at(3) ],
+
+		rowspanx(2)[ `M/G/1` ],
+		[ 1 ], [ 0.9 ], [ 11 ], [ 0.1 ],
+		[ #results_b.at(1).at(1) ],
+		[ #results_a.at(1).at(7) ],
+
+		[ 1 ], [ 0.99 ], [ 101 ], [ 0.01 ],
+		[ #results_b.at(1).at(2) ],
+		[ #results_a.at(1).at(11) ],
+	)),
+	kind: table,
+	caption: [Results]
+)
+
+#indent_par[We've included the workload variability from the previous exercise as the column $C^2$ to compare against. We can thus conclude that the variance and variability are correlated.]
+
+#indent_par[This makes sense, as despite our _elephants_ occurring less often, their larger size ensures that the users that come after them have a much higher average queue delay, which in turn increases the variance of the system.]
+
+#pagebreak()

zbuild.yaml

@@ -16,6 +16,7 @@ default:
   - rule: ex7_b
   - rule: ex7_c
   - rule: ex8_a
+  - rule: ex8_b
 
 rules:
   # Typst
@@ -217,3 +218,14 @@ rules:
     exec:
       - - Rscript
         - code/8a.R
+
+  # Exercise 8.b
+  ex8_b:
+    out:
+      - output/8b.csv
+    deps:
+      - code/8.R
+      - code/8b.R
+    exec:
+      - - Rscript
+        - code/8b.R