「AP過去問令和6年度春期午前問2」の版間の差分

2025年1月27日 (月) 17:36時点における最新版

問2(問題文)

　ATM(現金自動預払機)が1台ずつ設置してある二つの支店を統合し、統合後の支店にはATMを1台設置する。統合後のATMの平均待ち時間を求める式はどれか。ここで、待ち時間はM/M/1の待ち行列モデルに従い、平均待ち時間にはサービス時間を含まず、ATMを1台に統合しても十分に処理できるものとする。

〔条件〕

統合後の平均サービス時間：Ts

統合前のシステムの利用率：両支店ともρ

統合後の利用者数：統合前の両支店の利用者数の合計

ア　ρ1－ρ×Ts

イ　ρ1－2ρ×Ts

ウ　2ρ1－ρ×Ts

エ　2ρ1－2ρ×Ts

回答・解説

　M(Markovian:マルコフ指数分布)/M(Markovian:マルコフ指数分布)/1待ち行列モデルの問題の出題率は極めて高いので必ず覚えておくべきモデルです。

W q = ρ 1 - ρ \times T s

Wq：平均待ち時間

ρ：システムの利用率

Ts：平均サービス時間 ※μ=1Ts：サービス率でTsの代わりに1μとの積で表現する場合もあります。

　が覚えておくべき式です。Kendall(:ケンダル)の記法において到着間隔の分布の記号/サービス時間の分布記号/サーバ台数や窓口の数字という表記があって、分布記号にはMのマルコフ指数分布とGのGeneral(:任意)の分布とDのDeterministic(:一定)の分布の3種類が記述される可能性がありますが、情報処理技術者試験においてはM/M/1モデルしか出題されないので、記法やM or G or Dがあるというのは覚える必要はあまりないです。式を覚えておくことが大事です。

　統合後は単純に平均待ち時間が2倍になるだけというρ→2ρになる問題ですので１

W q = 2 ρ 1 - 2 ρ \times T s

が答えです。したがって

エ

　が答えです。　

AP過去問令和6年度春期午前問1前の問題へ

AP過去問令和6年度春期午前問3次の問題へ

AP過去問令和6年度春期午前問題に戻る

@@ 60行目: / 60行目: @@
 〔条件〕
-統合後の平均サービス時間：\( Ts \)
+統合後の平均サービス時間：<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_Ts" style="width: 1.383em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.076em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.281em, 1001.02em, 2.303em, -999.997em); top: -2.145em; left: 0em;"><span class="mrow"><span class="mi" style="font-family: NeoEulerMathJax_Main;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" style="font-family: NeoEulerMathJax_Main;">s</span></span><span style="display: inline-block; width: 0px; height: 2.15em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.062em; border-left: 0px solid; width: 0px; height: 1.042em;"></span></span></span>
-統合前のシステムの利用率：両支店とも\( \rho \)
+統合前のシステムの利用率：両支店とも<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_\rho" style="width: 0.718em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.565em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.536em, 1000.51em, 2.508em, -999.997em); top: -2.145em; left: 0em;"><span class="mrow"><span class="mi" style="font-family: NeoEulerMathJax_Main;">ρ</span></span><span style="display: inline-block; width: 0px; height: 2.15em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.321em; border-left: 0px solid; width: 0px; height: 0.977em;"></span></span></span>
 統合後の利用者数：統合前の両支店の利用者数の合計
@@ 80行目: / 80行目: @@
-$$ W_q = \frac{\rho}{1 - \rho} \times T_s $$
+<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" tabindex="0" style="text-align: center; position: relative;" role="presentation"><span class="none-math" id="Display:_W\_q_=_\frac{\rho}{1_-_\rho}_\times_T\_s" style="width: 9.359em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.365em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(0.821em, 1007.37em, 3.173em, -999.997em); top: -2.145em; left: 0em;"><span class="mrow"><span class="msubsup"><span style="display: inline-block; position: relative; width: 1.485em; height: 0px;"><span style="position: absolute; clip: rect(3.173em, 1001.08em, 4.144em, -999.997em); top: -3.986em; left: 0em;"><span class="mi" style="font-family: NeoEulerMathJax_Main;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.832em; left: 0.974em;"><span class="mi" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">q</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.258em;">=</span><span class="mfrac" style="padding-left: 0.258em;"><span style="display: inline-block; position: relative; width: 2.355em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.377em, 1000.51em, 4.349em, -999.997em); top: -4.65em; left: 50%; margin-left: -0.253em;"><span class="mi" style="font-family: NeoEulerMathJax_Main;">ρ</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.121em, 1002.15em, 4.349em, -999.997em); top: -3.321em; left: 50%; margin-left: -1.122em;"><span class="mrow"><span class="mn" style="font-family: NeoEulerMathJax_Main;">1</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">−</span><span class="mi" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">ρ</span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(0.872em, 1002.36em, 1.23em, -999.997em); top: -1.276em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.355em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.076em;"></span></span></span></span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">×</span><span class="msubsup" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.872em; height: 0px;"><span style="position: absolute; clip: rect(3.121em, 1000.57em, 4.144em, -999.997em); top: -3.986em; left: 0em;"><span class="mi" style="font-family: NeoEulerMathJax_Main;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.832em; left: 0.514em;"><span class="mi" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">s</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.15em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.166em; border-left: 0px solid; width: 0px; height: 2.731em;"></span></span></span></div>
-\( W_q \)：平均待ち時間
+<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_W\_q" style="width: 1.894em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.485em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.281em, 1001.49em, 2.559em, -999.997em); top: -2.094em; left: 0em;"><span class="mrow"><span class="msubsup"><span style="display: inline-block; position: relative; width: 1.485em; height: 0px;"><span style="position: absolute; clip: rect(3.173em, 1001.08em, 4.144em, -999.997em); top: -3.986em; left: 0em;"><span class="mi" style="font-family: NeoEulerMathJax_Main;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.832em; left: 0.974em;"><span class="mi" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">q</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.099em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.451em; border-left: 0px solid; width: 0px; height: 1.432em;"></span></span></span>：平均待ち時間
-\( \rho \)：システムの利用率
+<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_\rho" style="width: 0.718em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.565em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.536em, 1000.51em, 2.508em, -999.997em); top: -2.145em; left: 0em;"><span class="mrow"><span class="mi" style="font-family: NeoEulerMathJax_Main;">ρ</span></span><span style="display: inline-block; width: 0px; height: 2.15em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.321em; border-left: 0px solid; width: 0px; height: 0.977em;"></span></span></span>：システムの利用率
-\( T_s \)：平均サービス時間 ※\( \mu = \frac{1}{T_s} \)：サービス率で\( T_s \)の代わりに\( \frac{1}{\mu} \)との積で表現する場合もあります。
+<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_T\_s" style="width: 1.127em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.872em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.23em, 1000.87em, 2.406em, -999.997em); top: -2.094em; left: 0em;"><span class="mrow"><span class="msubsup"><span style="display: inline-block; position: relative; width: 0.872em; height: 0px;"><span style="position: absolute; clip: rect(3.121em, 1000.57em, 4.144em, -999.997em); top: -3.986em; left: 0em;"><span class="mi" style="font-family: NeoEulerMathJax_Main;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.832em; left: 0.514em;"><span class="mi" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">s</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.099em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.256em; border-left: 0px solid; width: 0px; height: 1.237em;"></span></span></span>：平均サービス時間 ※<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_\mu_=_\frac{1}{T\_s}" style="width: 3.633em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.866em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.076em, 1002.87em, 2.815em, -999.997em); top: -2.145em; left: 0em;"><span class="mrow"><span class="mi" style="font-family: NeoEulerMathJax_Main;">μ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.258em;">=</span><span class="mfrac" style="padding-left: 0.258em;"><span style="display: inline-block; position: relative; width: 0.77em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.326em, 1000.21em, 4.144em, -999.997em); top: -4.395em; left: 50%; margin-left: -0.151em;"><span class="mn" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.326em, 1000.62em, 4.246em, -999.997em); top: -3.577em; left: 50%; margin-left: -0.304em;"><span class="msubsup"><span style="display: inline-block; position: relative; width: 0.616em; height: 0px;"><span style="position: absolute; clip: rect(3.326em, 1000.41em, 4.144em, -999.997em); top: -3.986em; left: 0em;"><span class="mi" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.883em; left: 0.36em;"><span class="mi" style="font-size: 50%; font-family: NeoEulerMathJax_Main;">s</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(0.872em, 1000.77em, 1.23em, -999.997em); top: -1.276em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.77em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.076em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.15em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.711em; border-left: 0px solid; width: 0px; height: 1.951em;"></span></span></span>：サービス率で<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_T\_s" style="width: 1.127em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.872em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.23em, 1000.87em, 2.406em, -999.997em); top: -2.094em; left: 0em;"><span class="mrow"><span class="msubsup"><span style="display: inline-block; position: relative; width: 0.872em; height: 0px;"><span style="position: absolute; clip: rect(3.121em, 1000.57em, 4.144em, -999.997em); top: -3.986em; left: 0em;"><span class="mi" style="font-family: NeoEulerMathJax_Main;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.832em; left: 0.514em;"><span class="mi" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">s</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.099em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.256em; border-left: 0px solid; width: 0px; height: 1.237em;"></span></span></span>の代わりに<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_\frac{1}{\mu}" style="width: 0.974em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.77em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.025em, 1000.77em, 2.712em, -999.997em); top: -2.094em; left: 0em;"><span class="mrow"><span class="mfrac"><span style="display: inline-block; position: relative; width: 0.565em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.326em, 1000.21em, 4.144em, -999.997em); top: -4.395em; left: 50%; margin-left: -0.151em;"><span class="mn" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.531em, 1000.46em, 4.297em, -999.997em); top: -3.628em; left: 50%; margin-left: -0.202em;"><span class="mi" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">μ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(0.872em, 1000.57em, 1.23em, -999.997em); top: -1.276em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.565em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.076em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.099em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.646em; border-left: 0px solid; width: 0px; height: 1.886em;"></span></span></span>との積で表現する場合もあります。
@@ 93行目: / 93行目: @@
-　統合後は単純に平均待ち時間が2倍になるだけという\( \rho \)→\( 2\rho \)になる問題ですので１
+　統合後は単純に平均待ち時間が2倍になるだけという<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_\rho" style="width: 0.718em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.565em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.536em, 1000.51em, 2.508em, -999.997em); top: -2.145em; left: 0em;"><span class="mrow"><span class="mi" style="font-family: NeoEulerMathJax_Main;">ρ</span></span><span style="display: inline-block; width: 0px; height: 2.15em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.321em; border-left: 0px solid; width: 0px; height: 0.977em;"></span></span></span>→<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_2\rho" style="width: 1.332em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.025em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.281em, 1000.97em, 2.508em, -999.997em); top: -2.145em; left: 0em;"><span class="mrow"><span class="mn" style="font-family: NeoEulerMathJax_Main;">2</span><span class="mi" style="font-family: NeoEulerMathJax_Main;">ρ</span></span><span style="display: inline-block; width: 0px; height: 2.15em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.321em; border-left: 0px solid; width: 0px; height: 1.302em;"></span></span></span>になる問題ですので１
@@ 104行目: / 104行目: @@
 <script></yjavascript>
 <div class="imadake-left" style="transform-origin: top left;transform: scale(0.8);">
-$$ W_q = \frac{2\rho}{1 - 2\rho} \times T_s $$
+<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" tabindex="0" style="text-align: center; position: relative;" role="presentation"><span class="none-math" id="Display:_W\_q_=_\frac{2\rho}{1_-_2\rho}_\times_T\_s" style="width: 9.973em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.825em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(0.616em, 1007.83em, 3.173em, -999.997em); top: -2.145em; left: 0em;"><span class="mrow"><span class="msubsup"><span style="display: inline-block; position: relative; width: 1.485em; height: 0px;"><span style="position: absolute; clip: rect(3.173em, 1001.08em, 4.144em, -999.997em); top: -3.986em; left: 0em;"><span class="mi" style="font-family: NeoEulerMathJax_Main;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.832em; left: 0.974em;"><span class="mi" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">q</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.258em;">=</span><span class="mfrac" style="padding-left: 0.258em;"><span style="display: inline-block; position: relative; width: 2.815em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.121em, 1001.02em, 4.349em, -999.997em); top: -4.65em; left: 50%; margin-left: -0.509em;"><span class="mrow"><span class="mn" style="font-family: NeoEulerMathJax_Main;">2</span><span class="mi" style="font-family: NeoEulerMathJax_Main;">ρ</span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.121em, 1002.66em, 4.349em, -999.997em); top: -3.321em; left: 50%; margin-left: -1.378em;"><span class="mrow"><span class="mn" style="font-family: NeoEulerMathJax_Main;">1</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">−</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">2</span><span class="mi" style="font-family: NeoEulerMathJax_Main;">ρ</span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(0.872em, 1002.82em, 1.23em, -999.997em); top: -1.276em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.815em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.076em;"></span></span></span></span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">×</span><span class="msubsup" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.872em; height: 0px;"><span style="position: absolute; clip: rect(3.121em, 1000.57em, 4.144em, -999.997em); top: -3.986em; left: 0em;"><span class="mi" style="font-family: NeoEulerMathJax_Main;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.832em; left: 0.514em;"><span class="mi" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">s</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.15em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.166em; border-left: 0px solid; width: 0px; height: 2.99em;"></span></span></span></div>
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