Started work on exercise 8c.

code/8c.R

@@ -1,23 +1,36 @@
-ρ <- 0.5
-s1 <- 1
-p1 <- 0.99
-s2 <- 101
-p2 <- 0.01
+params_all <- list(
+  list(s1 = 1, p1 = 0.9, s2 = 11, p2 = 0.1),
+  list(s1 = 1, p1 = 0.99, s2 = 101, p2 = 0.01)
+)
 
-μ <- 1 / (s1 * p1 + s2 * p2)
-λ <- ρ * μ
-λ1 <- p1 * λ
-λ2 <- p2 * λ
+calc_stats <- function(params) {
+  ρ <- 0.5
+  μ <- 1 / (params$s1 * params$p1 + params$s2 * params$p2)
+  λ <- ρ * μ
+  λ1 <- params$p1 * λ
+  λ2 <- params$p2 * λ
 
-wq1 <- (λ1 * s1^2 + λ2 * s2^2) / (2 * (1 - λ1 * s1))
-wq2 <- (λ1 * s1^2 + λ2 * s2^2) / (2 * (1 - λ1 * s1) * (1 - λ1 * s1 - λ2 * s2))
+  wq1 <- (λ1 * params$s1^2 + λ2 * params$s2^2) / (2 * (1 - λ1 * params$s1))
+  wq2 <- (λ1 * params$s1^2 + λ2 * params$s2^2) / (2 * (1 - λ1 * params$s1) * (1 - λ1 * params$s1 - λ2 * params$s2))
 
-wq <- (λ1 * wq1 + λ2 * wq2) / (λ1 + λ2)
+  wq <- (λ1 * wq1 + λ2 * wq2) / (λ1 + λ2)
 
-cat(sprintf("μ = %.2f\n", μ))
-cat(sprintf("λ = %.2f\n", λ))
-cat(sprintf("λ1 = %.4f\n", λ1))
-cat(sprintf("λ2 = %.4f\n", λ2))
-cat(sprintf("wq1 = %.2f\n", wq1))
-cat(sprintf("wq2 = %.2f\n", wq2))
-cat(sprintf("wq = %.2f\n", wq))
+  data.frame(
+    s1 = signif(params$s1, 4),
+    p1 = signif(params$p1, 4),
+    s2 = signif(params$s2, 4),
+    p2 = signif(params$p2, 4),
+    μ = signif(μ, 4),
+    λ = signif(λ, 4),
+    λ1 = signif(λ1, 4),
+    λ2 = signif(λ2, 4),
+    wq1 = signif(wq1, 4),
+    wq2 = signif(wq2, 4),
+    wq = signif(wq, 4)
+  )
+}
+
+data <- lapply(params_all, calc_stats)
+data <- do.call(rbind, data)
+str(data)
+write.table(data, "output/8c.csv", sep = "\t", row.names = FALSE)

typst/exercises/8.typ

@@ -103,3 +103,65 @@
 #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()
+
+==== c.
+
+#let results_c = csv("/output/8c.csv", delimiter: "\t")
+
+#indent_par[In our current implementation, we treat both the _elephants_ and _mice_ the same. _Elephants_ will always have a large size and thus need more time in queue, but this shouldn't affect the _mice_ that can be dispatched quickly.]
+
+#indent_par[To remedy this, we can treat both categories of users differently, by performing *packet scheduling*. In specific, we can use a *strict priority* with _mice_ having a higher priority than the _elephants_. This leads to a higher average delay for the _elephants_, but with the tradeoff of the _mice_ having much lower average delay.]
+
+#indent_par[To be able to calculate exactly whether this tradeoff is valuable, we can use the following formulas 11 through 15 to calculate the average queuing delay for each type of user:]
+
+$ λ_1 = p_1 dot λ $
+$ λ_2 = p_2 dot λ $
+
+$ W_"q1" = (λ_1 s_1^2 + λ_2 s_2^2) / (2 (1 - λ_1 s_1)) $
+$ W_"q2" = (λ_1 s_1^2 + λ_2 s_2^2) / (2 (1 - λ_1 s_1) (1 - λ_1 s_1 - λ_2 s_2)) $
+
+$ W_q = (λ_1 W_"q1" + λ_2 W_"q2") / (λ_1 + λ_2) $
+
+#indent_par[Where $W_"q1"$ is the average queueing delay of _mice_, $W_"q2"$ is the average queueing delay of _elephants_ and $W_q$ is the total average queueing delay]
+
+#indent_par[With these formulas in hand, we have computed the values in the following table 17:]
+
+#figure(
+	pad(1em, tablex(
+		columns: (auto, auto, auto, auto, auto, auto, 1fr, 1fr, 1fr, 1fr),
+		align: center + horizon,
+
+		rowspanx(3)[ System ],
+		rowspanx(3)[ $ρ$ ],
+		rowspanx(3)[ $S_1$ ],
+		rowspanx(3)[ $p_1$ ],
+		rowspanx(3)[ $S_2$ ],
+		rowspanx(3)[ $p_2$ ],
+		colspanx(4)[ Average delay ],
+		rowspanx(2)[ Standard ],
+		colspanx(3)[ Strict priority packet scheduling ],
+		[ _mice_ ],
+		[ _elephants_ ],
+		[ Total ],
+
+		rowspanx(2)[ `M/G/1` ],
+		rowspanx(2)[ 0.5 ],
+		[ 1 ], [ 0.9 ], [ 11 ], [ 0.1 ],
+		[ #results_a.at(1).at(6) ],
+		[ #results_c.at(1).at(8) ],
+		[ #results_c.at(1).at(9) ],
+		[ #results_c.at(1).at(10) ],
+
+		[ 1 ], [ 0.99 ], [ 101 ], [ 0.01 ],
+		[ #results_a.at(1).at(10) ],
+		[ #results_c.at(2).at(8) ],
+		[ #results_c.at(2).at(9) ],
+		[ #results_c.at(2).at(10) ],
+	)),
+	kind: table,
+	caption: [Results]
+)
+
+#indent_par[The average delay of the _mice_ is now considerably lower than before. The _elephants_' average delay rose about as much as the _mice_'s lowered, but given that we have many more _mice_ than _elephants_, this is a valuable tradeoff, which can be further reinforced by looking at the average total, which is lower in general, by a considerable amount.]
+
+#pagebreak()

zbuild.yaml

@@ -17,6 +17,7 @@ default:
   - rule: ex7_c
   - rule: ex8_a
   - rule: ex8_b
+  - rule: ex8_c
 
 rules:
   # Typst
@@ -229,3 +230,14 @@ rules:
     exec:
       - - Rscript
         - code/8b.R
+
+  # Exercise 8.c
+  ex8_c:
+    out:
+      - output/8c.csv
+    deps:
+      - code/8.R
+      - code/8c.R
+    exec:
+      - - Rscript
+        - code/8c.R