From 257a4d27f5d3c4557af98bc210ab2332ecddc5ba Mon Sep 17 00:00:00 2001 From: Luca Lombardo Date: Tue, 21 Mar 2023 22:44:05 +0100 Subject: [PATCH] gitignore --- .gitignore | 301 +++++++ discorso/0uFWo7C.png | Bin 0 -> 93877 bytes discorso/3zlg6uj.png | Bin 0 -> 100663 bytes discorso/9LBRzWR.png | Bin 0 -> 77864 bytes discorso/CBOSr9C.png | Bin 0 -> 70811 bytes discorso/E9pbGl1.png | Bin 0 -> 40442 bytes discorso/EZp90oP.png | Bin 0 -> 41585 bytes discorso/FcngTm3.png | Bin 0 -> 78100 bytes discorso/GFF6s3y.png | Bin 0 -> 45823 bytes discorso/GYJt39w.png | Bin 0 -> 45390 bytes discorso/R1VGHAR.png | Bin 0 -> 76875 bytes discorso/T9RIVqa.png | Bin 0 -> 90799 bytes discorso/Tcjl81t.png | Bin 0 -> 37371 bytes discorso/VrxLKYI.png | Bin 0 -> 46268 bytes discorso/avNOocE.png | Bin 0 -> 40315 bytes discorso/gtHnIzQ.png | Bin 0 -> 33824 bytes discorso/hVQg024.png | Bin 0 -> 53949 bytes discorso/lLnSEzP.png | Bin 0 -> 21877 bytes discorso/main.pdf | Bin 0 -> 1092518 bytes discorso/main.tex | 519 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slide/main.tex | 961 ++++++++++++++++++++++ slide/pSUZiNk.png | Bin 0 -> 44525 bytes slide/px6kcg9.png | Bin 0 -> 179091 bytes slide/qUQVrEZ.png | Bin 0 -> 103362 bytes 47 files changed, 1781 insertions(+) create mode 100644 .gitignore create mode 100644 discorso/0uFWo7C.png create mode 100644 discorso/3zlg6uj.png create mode 100644 discorso/9LBRzWR.png create mode 100644 discorso/CBOSr9C.png create mode 100644 discorso/E9pbGl1.png create mode 100644 discorso/EZp90oP.png create mode 100644 discorso/FcngTm3.png create mode 100644 discorso/GFF6s3y.png create mode 100644 discorso/GYJt39w.png create mode 100644 discorso/R1VGHAR.png create mode 100644 discorso/T9RIVqa.png create mode 100644 discorso/Tcjl81t.png create mode 100644 discorso/VrxLKYI.png create mode 100644 discorso/avNOocE.png create mode 100644 discorso/gtHnIzQ.png create mode 100644 discorso/hVQg024.png create mode 100644 discorso/lLnSEzP.png create mode 100644 discorso/main.pdf create mode 100644 discorso/main.tex create 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slide/qUQVrEZ.png diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..e964244 --- /dev/null +++ b/.gitignore @@ -0,0 +1,301 @@ +## Core latex/pdflatex auxiliary files: +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc +*.fmt +*.fot +*.cb +*.cb2 +.*.lb + +## Intermediate documents: +*.dvi +*.xdv +*-converted-to.* +# these rules might exclude image files for figures etc. +# *.ps +# *.eps +# *.pdf + +## Generated if empty string is given at "Please type another file name for output:" +.pdf + +## Bibliography auxiliary files (bibtex/biblatex/biber): +*.bbl +*.bcf +*.blg +*-blx.aux +*-blx.bib +*.run.xml + +## Build tool auxiliary files: +*.fdb_latexmk +*.synctex +*.synctex(busy) +*.synctex.gz +*.synctex.gz(busy) +*.pdfsync + +## Build tool directories for auxiliary files +# latexrun +latex.out/ + +## Auxiliary and intermediate files from other packages: +# algorithms +*.alg +*.loa + +# achemso +acs-*.bib + +# amsthm +*.thm + +# beamer +*.nav +*.pre +*.snm +*.vrb + +# changes +*.soc + +# comment +*.cut + +# cprotect +*.cpt + +# elsarticle (documentclass of Elsevier journals) +*.spl + +# endnotes +*.ent + +# fixme +*.lox + +# feynmf/feynmp +*.mf +*.mp +*.t[1-9] +*.t[1-9][0-9] +*.tfm + +#(r)(e)ledmac/(r)(e)ledpar +*.end +*.?end +*.[1-9] +*.[1-9][0-9] +*.[1-9][0-9][0-9] +*.[1-9]R +*.[1-9][0-9]R +*.[1-9][0-9][0-9]R +*.eledsec[1-9] +*.eledsec[1-9]R +*.eledsec[1-9][0-9] +*.eledsec[1-9][0-9]R +*.eledsec[1-9][0-9][0-9] +*.eledsec[1-9][0-9][0-9]R + +# glossaries +*.acn +*.acr +*.glg +*.glo +*.gls +*.glsdefs +*.lzo +*.lzs +*.slg +*.slo +*.sls + +# uncomment this for glossaries-extra (will ignore makeindex's style files!) +# *.ist + +# gnuplot +*.gnuplot +*.table + +# gnuplottex +*-gnuplottex-* + +# gregoriotex +*.gaux +*.glog +*.gtex + +# htlatex +*.4ct +*.4tc +*.idv +*.lg +*.trc +*.xref + +# hyperref +*.brf + +# knitr +*-concordance.tex +# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files +# *.tikz +*-tikzDictionary + +# listings +*.lol + +# luatexja-ruby +*.ltjruby + +# makeidx +*.idx +*.ilg +*.ind + +# minitoc +*.maf +*.mlf +*.mlt +*.mtc[0-9]* +*.slf[0-9]* +*.slt[0-9]* +*.stc[0-9]* + +# minted +_minted* +*.pyg + +# morewrites +*.mw + +# newpax +*.newpax + +# nomencl +*.nlg +*.nlo +*.nls + +# pax +*.pax + +# pdfpcnotes +*.pdfpc + +# sagetex +*.sagetex.sage +*.sagetex.py +*.sagetex.scmd + +# scrwfile +*.wrt + +# svg +svg-inkscape/ + +# sympy +*.sout +*.sympy +sympy-plots-for-*.tex/ + +# pdfcomment +*.upa +*.upb + +# pythontex +*.pytxcode +pythontex-files-*/ + +# tcolorbox +*.listing + +# thmtools +*.loe + +# TikZ & PGF +*.dpth +*.md5 +*.auxlock + +# titletoc +*.ptc + +# todonotes +*.tdo + +# vhistory +*.hst +*.ver + +# easy-todo +*.lod + +# xcolor +*.xcp + +# xmpincl +*.xmpi + +# xindy +*.xdy + +# xypic precompiled matrices and outlines +*.xyc +*.xyd + +# endfloat +*.ttt +*.fff + +# Latexian +TSWLatexianTemp* + +## Editors: +# WinEdt +*.bak +*.sav + +# Texpad +.texpadtmp + +# LyX +*.lyx~ + +# Kile +*.backup + +# gummi +.*.swp + +# KBibTeX +*~[0-9]* + +# TeXnicCenter +*.tps + +# auto folder when using emacs and auctex +./auto/* +*.el + +# expex forward references with \gathertags +*-tags.tex + +# standalone packages +*.sta + +# Makeindex log files +*.lpz + +# xwatermark package +*.xwm + +# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib +# option is specified. 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+ \begin{figure}[h!] + \centering + \includegraphics[width=0.6\textwidth]{lLnSEzP.png} + \end{figure} + Usando la regole del mixed product per il prodotto di Kronecker, ed usando \ref{eq:invariant} si verifica che la soluzione del sistema di equazioni lineari è + \begin{equation} + y = x \bigotimes \delta + \end{equation} + dove $\delta$ è come definita in \ref{eq:invariant} ed $x$ è la soluzione del sistema + \begin{equation}\label{eq:system} + xS = 0, \qquad xe = 1 + \end{equation} + per sostituzione diretta, si verifica facilmente che le componenti del vettore $x = (x_0, x_1, ... , x_L)$, corrispondenti alle uniche soluzioni del sistema \ref{eq:system}, sono date da + \begin{equation}\label{eq:xi} + x_0 = \frac{\mu_2}{L(1-q)\mu_1 + \mu_2}, \qquad x_i = \frac{\mu_1(1-q)}{L(1-q)\mu_1 + \mu_2}, \qquad i = 1, ..., L + \end{equation} + La tesi segue delle equazioni \ref{eq:ergodicity}, \ref{eq:bigotimes} e \ref{eq:xi} assieme alla definizione di $\lambda$. + \end{proof} + +\end{teorema} + +\begin{osservazione} + La condizione di stabilità data nell'Equazione \ref{eq:ergodicity} può essere intuitivamente spiegata nel seguente modo. In generale, la condizione di ergodicità richiede che il tasso di arrivo dei clienti per unità di tempo debba essere inferiore al tasso di servizio che i clienti ricevono per unità di tempo quando il sistema è sovraccarico (nel senso che il numero di clienti presenti nel sistema è molto grande). Qui, il tasso di arrivo dei clienti è $\lambda$ per unità di tempo. Il tasso di servizio dei clienti quando il sistema è sovraccarico è la somma di $\mu_1$ (il tasso di servizio per unità di tempo fornito dal server principale) e il tasso di servizio (per unità di tempo) fornito dal server secondario. Quest'ultimo tasso di servizio è $0$ quando il server secondario non è presente nel sistema, il che avviene con probabilità $x_0$. Quando il server secondario è presente nel sistema, che avviene con probabilità $(1 - x_0)$, i clienti ricevono il servizio e lasciano il sistema ad un tasso di $\mu_2 (1 - \nu)$ per unità di tempo. Pertanto, il tasso di servizio medio totale è dato da: + \begin{equation} + \mu = \mu_1 + \mu_2(1 - \nu) \frac{L(1-q)\mu_1}{L(1-q)\mu_1 + \mu_2} + \end{equation} + da cui segue che la condizione di ergodicità vista in \ref{eq:ergodicity} +\end{osservazione} + +\begin{osservazione} + La probabilità, $x_0$, che il secondo server non sia presente nel sistema in un qualsiasi momento in cui il sistema è sovraccarico può essere facilmente calcolata dalla seguente considerazione. Si considerino i periodi in cui il server secondario non è presente nel sistema (ovvero, la durata media di questo periodo è $\frac{1}{\rho \mu_1}$) alternati ai periodi in cui il server secondario è presente nel sistema. Quando il sistema attiva un server secondario (quando il sistema è sovraccarico, il server secondario viene assegnato a gestire $L$ per i servizi), la durata media del server secondario continuamente presente nel sistema è data da $\frac{L}{\mu_2}$. Pertanto, abbiamo: + \begin{equation} + x_0 = \frac{\frac{1}{\mu_1(1-q)}}{\frac{1}{\mu_1(1-q)} + \frac{L}{\mu_2}} = \frac{\mu_2}{L(1-q)\mu_1 + \mu_2} + \end{equation} + che corrisponde all'espressione \ref{eq:xi} vista in precedenza. +\end{osservazione} + +\subsection{Computation of the Performance Measures of the System} + +Sotto l'assunzione che la condizione di ergodicità data dalla relazione \ref{eq:ergodicity} sia valida, esistono le seguenti probabilità stazionarie degli stati del CTMC $\zeta_t, t \geq 0$: +\begin{equation} + \pi(i,n,\zeta) = \lim_{t \to \infty} P\{i_t = i, n_t = n, \zeta_t = \zeta\}, ~ i \geq 0, \quad n \in \{0, 1, \dots , \min\{i,L\}\}, \quad \zeta \in \{0, \dots, n\} +\end{equation} +Consideriamo i vettori riga delle probabilità di stato stazionario $\pi_i$ come segue: il vettore riga $\pi(i,n)$ è dato da $\pi(i,n) = (\pi(i,n,1), ..., \pi(i,n,m))$ e +\begin{equation} + \pi_i = (\pi(i,0), ..., \pi(i, \min \{i,L\})), \quad i \geq 0 +\end{equation} +È ben noto che i vettori di probabilità stazionari $\pi_i, i \geq 0$, soddisfano il sistema di equazioni algebriche lineari (equazioni di equilibrio): +\begin{equation} + (\pi_0, \pi_1, \pi_, \dots)Q = 0 \qquad (\pi_0, \pi_1, \pi_, \dots)e = 1 +\end{equation} +dove $Q$ è la matrice di transizione del CTMC $\zeta_t, t \geq 0$ e $e$ è il vettore colonna di tutti gli elementi $1$. La soluzione del problema di calcolo della distribuzione stazionaria di una QBD indipendente dal livello è ben nota; si veda [37]. Per i livelli in cui le transizioni della QBD non dipendono dal livello, i vettori di probabilità stazionaria sono trovati in forma matriciale geometrica. I vettori di probabilità stazionaria dei livelli di confine, in cui le transizioni della QBD dipendono dal livello, sono quindi direttamente trovati come soluzione del sistema di equazioni algebriche lineari. Tuttavia, se il numero di livelli di confine è grande (cosa che accade nel nostro modello se $L$ è grande), questo sistema ha una grande dimensione. Qui presentiamo un algoritmo che sfrutta essenzialmente la struttura tridiagonale a blocchi ma dipendente dal livello del generatore per i livelli minori di $L + 1$. L'algoritmo utilizzato per risolvere il sistema infinito di equazioni di equilibrio è presentato nella seguente teorema: + +\begin{teorema} + I vettori $\pi_i, i \geq 0$, sono trovati come soluzione del sistema di equazioni algebriche lineari: + \begin{equation} + \pi_i = \alpha_i \big( \sum_{l=0}^\infty \alpha_l e \big)^{-1}, \qquad i \geq 0 + \end{equation} + dove il vettore $\alpha_0$ è calcolato come l'unica soluzione del sistema di equazioni + \begin{equation} + \alpha_0(Q_{0,0} + Q_{0,1}G_0) = 0, \qquad \alpha_0e = 1 + \end{equation} + ed i vettori $\alpha_i, i \geq 1$, sono definiti come + \begin{equation} + \alpha_i = \alpha_0 \prod_{l=1}^i R_l, \qquad i \geq 1 + \end{equation} + o tramite la formula ricorsiva + \begin{equation}\label{eq:rec-alpha} + \alpha_i = \alpha_{i-1} R_i, \qquad i \geq 1 + \end{equation} + dove + \begin{equation} + R = + \begin{cases} + -Q_{i-1,i}(Q_{i,i} + Q_{i,i+1}G_i)^{-1}Q & 1 \leq i \leq L-1 \\ + -Q_{L-1,L}(Q_{L,L} + Q^+G)^{-1} & i = L \\ + -Q^+(Q^0 + Q^+G)^{-1} = R & i>L + \end{cases} + \end{equation} + Le matrici stocastiche $G_i$ sono calcolate utilizzando la seguente formula ricorsiva all'indietro: + \begin{equation} + \begin{split} + G_L &= G \\ + G_L-1 &= -(Q_{L,L} + Q^+G_L)^{-1}Q_{L,L-1} \\ + G_i &= -(Q_{i+1,i+1} + Q_{i+1,i+2}G_{i+1})^{-1}Q_{i+1,i}, \qquad i = L-2, L-3, \dots, 0 + \end{split} + \end{equation} + dove la matrice $G$ è la minima soluzione non negativa dell'equazione quadratica matriciale + \begin{equation} + Q^+G^2 + Q^0G + Q^- = 0 + \end{equation} + + \begin{proof} + non fornita + \end{proof} +\end{teorema} + +Questo algoritmo è una modifica efficace dell'algoritmo per il calcolo della distribuzione stazionaria del CTMC asintoticamente quasi-Toeplitz (vedi, ad esempio, [31], pp. 145-146). In [31], i vettori $\pi_i$ sono calcolati come $\pi_i = \pi_0 F_i , i \geq 0$, dove le matrici $F_i$ sono ottenute dalla ricorsione di matrici simile all'Equazione \ref{eq:rec-alpha}. Utilizzando la ricorsione di vettori come indicato nell'Equazione \ref{eq:rec-alpha} invece della ricorsione di matrici corrispondente, si ottiene una significativa riduzione della memoria del computer richiesta e del tempo di esecuzione. L'esistenza delle inverse delle matrici (tutte sub-generatori irriducibili) che appaiono nell'algoritmo sopra segue immediatamente, ad esempio, dal teorema di O. Tausska [55]. Inoltre, queste matrici sono semi-stabili (e quindi le inverse dei negativi di queste matrici sono non negative), risultando nella produzione di procedure ricorsive stabili nell'implementazione numerica dell'algoritmo. + +\begin{corollario} + Per $i \geq L$ vale la seguente formula + \begin{equation} + \alpha_i = \alpha_L R^{i-L} + \end{equation} + dove + \begin{equation} + \alpha_L = \alpha_0 \prod_{l=1}^L R_l + \end{equation} +\end{corollario} + +\subsection{Computation of the Performance Measures of the System} +Per studiare il modello di coda in questione qualitativamente e confrontarlo con la corrispondente coda classica MAP/M/1 per valutare l'impatto del processo di reclutamento, dobbiamo sviluppare alcune misure di performance chiave. Di seguito sono elencate alcune di queste insieme alle loro formule: \\ + +\textsc{Guarda paper, troppo lungo} + +\clearpage + +\section{GI/M/1 Approach} + +In questa sezione, presentiamo brevemente come analizzare il sistema di coda in studio utilizzando code di tipo GI/M/1 a tempo continuo. Tenendo traccia del numero di clienti in attesa nella coda insieme allo stato del server principale (occupato o libero) e allo stato del server secondario (non presente o presente con un numero specificato di clienti assegnati), possiamo studiare il modello come una CTMC di tipo GI/M/1 come segue. \vspace*{0.4cm} + +\noindent Definiamo come prima cosa lo spazio degli stati $\Omega$ del CTMC come: +\begin{equation} + \Omega = \{ (i,j,k) ~ : ~ i \geq 0, 0 \leq j \leq K, 1 \leq k \leq m \} +\end{equation} +In seguito, consideriamo $e_r$ come un vettore colonna con 1 nella posizione $r-esima$ e 0 altrove. Notare che quando necessario, indicheremo la dimensione tra parentesi. Ad esempio, $e(L + 1)$ indicherà un vettore colonna di 1 con dimensione $L+1$. La "T" che appare come pedice in un vettore o una matrice sta per la notazione di trasposizione. Quindi, $e^T$ indicherà un vettore riga di 1. \vspace*{0.4cm} + +\noindent Definiamo il livello $\textbf{i} = \{(i, j, k) : 0 \leq j \leq L, 1 \leq k \leq m\} = \{(\textbf{i}, 0), \dots, (\textbf{i}, L)\}, i \geq 0$. Notare che il livello $(\textbf{i},\textbf{j})$ indica che il server principale è occupato (a patto che $i>0$), ci sono $i-1$ clienti in attesa nella coda principale, il server secondario (a patto che $j>0$) è occupato e il processo di arrivo si trova in varie fasi. Il livello $(\textbf{0},\textbf{0})$ corrisponde al sistema inattivo con il processo MAP in una delle $m$ fasi. Il generatore $\widetilde{Q}$ della CTMC che governa il sistema in studio è dato da: +\begin{figure}[h!] + \centering + \includegraphics[width=0.9\textwidth]{FcngTm3.png} + \label{fig:GI-M-1} +\end{figure} + +\textsc{Guardare paper, troppo lungo} + +\section{Risultati numerici} + +In questa sezione, forniamo alcuni esempi illustrativi utilizzando cinque diversi processi di arrivo. Di questi cinque, tre sono processi di rinnovo e due sono processi correlati. In particolare, prendiamo i cinque MAP come: +\begin{itemize} + \item \textbf{ERL}: Questo è un Erlang di ordine 5 con parametro 2.5 in ciascuno dei 5 stati. Notare che qui abbiamo $\lambda = 0.5, \sigma = 0.899427$ e $\rho_c = 0$. + \item \textbf{EXP}: Questo è un esponenziale con una frequenza di 0.5. Notare che qui abbiamo $\lambda = 0.5, \sigma = 2$ e $\rho_c = 0$. + \item \textbf{HEX}: Questa è una distribuzione iper-esponenziale con una probabilità di mixing data da (0.5, 0.3, 0.15, 0.04, 0.01) con i corrispondenti tassi della distribuzione esponenziale pari a (1.09, 0.545, 0.2725, 0.13625, 0.068125). Qui abbiamo $\lambda = 0.5, \sigma = 3.3942$ e $\rho_c = 0$. +\end{itemize} +I due processi correlati, negativo e positivo, sono i seguenti: +\begin{itemize} + \item \textbf{NCR}: Questo è un MAP negativamente correlato, con matrici di rappresentazione date da: + \begin{figure}[h!] + \centering + \includegraphics[width=0.6\textwidth]{GYJt39w.png} + \label{fig:NCR} + \end{figure} + dove abbiamo $\lambda = 0.5, \sigma = 2.02454$ e $\rho_c = -0.57855$. + \item \textbf{PCR}: Questo è un MAP positivamente correlato, con matrici di rappresentazione date da: + \begin{figure} + \centering + \includegraphics[width=0.6\textwidth]{GFF6s3y.png} + \label{fig:PCR} + \end{figure} + dove abbiamo $\lambda = 0.5, \sigma = 2.02454$ e $\rho_c = 0.57855$. +\end{itemize} +Guardando le cinque MAP sopra riportate, è evidente che sono tutte qualitativamente diverse. È importante sottolineare che il processo di arrivo denominato PCR è ideale per situazioni in cui gli arrivi dei clienti sono altamente irregolari, con periodi alternati di congestione e di scarsità del sistema. Tali arrivi sono comuni nella pratica, specialmente nelle telecomunicazioni e nelle industrie dei servizi. È importante notare, inoltre, che il processo di arrivo denominato HEX è noto per presentare un comportamento irregolare simile nel senso che gli arrivi con tempi tra di essi più brevi sono separati da tempi più lunghi. Tuttavia, la differenza tra questi due processi sta nella correlazione positiva presente nel processo PCR. L'impatto di questa correlazione positiva, così come dell'elevata variabilità nei tempi tra gli arrivi, come nei due processi sopra citati, è stato ben documentato in letteratura (vedi, ad esempio, riferimenti [29,30]). Discutiamo tre esempi numerici rappresentativi e illustrativi per evidenziare la natura qualitativa del modello in studio. + +\subsection{Primo esempio numerico} +Qui discutiamo l'impatto del parametro L su alcune misure di performance del sistema selezionate per tutte e cinque le MAP. Innanzitutto, fissiamo $\mu_1=1$, $\mu_2=0.5$, $q=0.5$, e $\nu=0.4$, e variamo $L$ da 1 a 30. + +\begin{figure}[h] + \centering + \includegraphics[width=0.7\textwidth]{gtHnIzQ.png} + \caption{Impact of L on the average number of customers in the system $L_{system}$ for different MAPs.} + \label{fig:1} +\end{figure} +La Figura \ref{fig:1} illustra chiaramente l'effetto dell'irregolarità nel processo di arrivo, ovvero PCR. Il numero medio di clienti nel sistema nel caso di PCR è molte volte maggiore rispetto agli altri MAPs. Vale la pena sottolineare che per i primi quattro MAPs, la misura $L_\text{system}$ è una funzione non decrescente di $L$, mentre per PCR si osserva un trend non crescente. Ciò spiega il ruolo della correlazione, soprattutto positiva, e non dovrebbe essere ignorato. Inoltre, un valore elevato di L indica che quando un server secondario viene reclutato, verranno assegnati più clienti e, a causa della lentezza del server secondario (rispetto al server principale), c'è una alta probabilità, soprattutto per i casi dei primi quattro MAPs, che il sistema abbia in media più clienti nel sistema. Similmente a quanto noto nella coda classica, ovvero il numero medio nel sistema aumenta con l'aumento della variabilità nei tempi di arrivo tra gli arrivi di rinnovo, vediamo che questo comportamento si verifica anche qui nei primi tre MAPs, che corrispondono agli arrivi di rinnovo. \vspace{0.4cm} + +\noindent Tuttavia, per quanto riguarda gli arrivi PCR, osserviamo un trend interessante ma opposto, ovvero un trend decrescente. Questo può essere intuitivamente spiegato come segue. Innanzitutto, si osserva che il sistema L ha un valore massimo di 15.3983 quando $L = 1$, il che può essere spiegato utilizzando il fatto che, quando $L = 1$, i server secondari lasciano il sistema dopo aver servito un solo cliente; con una probabilità del solo 0.5 di essere reclutati, la coda tende ad accumularsi rapidamente. Aumentando L, i server secondari sono maggiormente coinvolti nella pulizia della coda, soprattutto quando gli arrivi avvengono a sprazzi, e quindi $L_\text{system}$ diminuisce. Raggiunge un valore minimo di 11.9757 quando $L = 16$ e poi inizia ad aumentare a causa della mancata possibilità di essere serviti dal server principale. Per $L = 30, L_\text{system} = 12.0605$ \vspace{0.4cm} + +\begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{Tcjl81t.png} + \caption{ Dependence of the average number of customers with the secondary server L sec on the + parameter L for different MAPs.} + \label{fig:2} +\end{figure} + +\noindent La Figura \ref{fig:2} mostra il comportamento della media del numero di clienti con il server secondario $L_{\mathrm{sec}}$. + +\clearpage +\begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{pSUZiNk.png} + \caption{ Dependence of the probability $P_{\mathrm{idle-system}}$ that the system is idle at an arbitrary moment on the parameter L for different MAPs} + \label{fig:3} +\end{figure} + +La Figura \ref{fig:3} illustra il comportamento della probabilità, $P_{\mathrm{idle-system}}$, che il sistema sia inattivo in un momento arbitrario. Questa figura corrisponde alla Figura \ref{fig:1} su due aspetti. Il primo è che mostra anche una grande differenza nella misura quando viene confrontata con vari MAPs. Quando si cerca di trovare un valore ottimale di L, è evidente che conta quale misura viene scelta come funzione obiettivo e il tipo di MAPs utilizzato quando tutti gli altri parametri sono fissati. Ad esempio, se si considera il processo di arrivo PCR, il valore ottimale di L è 16 se si cerca di minimizzare $L_{\mathrm{system}}$. Tuttavia, se la misura $P_{\mathrm{idle-system}}$ è l'obiettivo del problema di ottimizzazione, allora $L = $6 produce il valore più grande per questa misura. \vspace{0.4cm} + +\begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{VrxLKYI.png} + \caption{ Dependence of the probability $P_{\mathrm{idle-busy}}$ that the main server is idle while the secondary server is busy on the parameter L for different MAPs} + \label{fig:4} +\end{figure} + +\begin{figure} + \centering + \includegraphics[width=0.8\textwidth]{EZp90oP.png} + \caption{Dependence of the probability $P_{\mathrm{busy-idle}}$ that the main server is busy while the secondary server is idle on the parameter L for different MAPs.} + \label{fig:5} +\end{figure} + +Le Figure \ref{fig:4} e \ref{fig:5} illustrano il comportamento delle probabilità $P_{\mathrm{idle-busy}}$ e $P_{\mathrm{busy-idle}}$, che corrispondono rispettivamente al momento in cui il server principale è inattivo con il server secondario occupato, e al momento in cui il server principale è occupato con il server secondario inattivo, in un momento arbitrario. Mentre la prima probabilità aumenta all'aumentare di L, la seconda probabilità diminuisce. Da queste figure, si possono notare le differenze essenziali in queste probabilità in vari scenari. + +\clearpage + +\subsection{Secondo esempio numerico} +Lo scopo di questo esempio è di indagare l'impatto dei parametri $q$ (ricorda che questo è la probabilità che un cliente servito si rifiuti di agire come server secondario) e $\nu$ (questa è la probabilità che un cliente servito da un server secondario sia insoddisfatto e torni al sistema). Fissiamo il valore di $L$ a $10$ (punto medio tra i due valori ottimali menzionati nel primo esempio). Fissiamo anche i tassi di servizio come $\mu_1 = 1$ e $\mu_2 = 0.5$ e indaghiamo la dipendenza di diverse misure di prestazione dalle probabilità $q$ e $\nu$. Variamo i valori di queste probabilità da $0$ a $1$ con passo $0.05$. Si noti che il valore $q = 1$ corrisponde al classico sistema MAP/M/1 con il tasso di servizio $\mu_1$. \vspace{0.4cm}. + +\noindent In questo esempio ci concentriamo sul processo di arrivo etichettato come PCR, la cui scelta è basata sul comportamento di questo processo sulle misure evidenziato nel primo esempio illustrativo. Dalla Figura \ref{fig:6}, che mostra la dipendenza del numero medio di clienti nel sistema $L_\mathrm{system}$ dai parametri $q$ e $\nu$, deduciamo diverse osservazioni interessanti. + +\begin{figure}[h] + \centering + \includegraphics[width=0.7\textwidth]{qUQVrEZ.png} + \caption{Dependence of the average number of customers in the system $L_\mathrm{system}$ on the parameters q and $\nu$.} + \label{fig:6} +\end{figure} + +Il valore di $L_\mathrm{system}$ è minimo con un valore di $7,9328$ quando il cliente servito è sempre disponibile per essere reclutato (quando il sistema ne ha bisogno) e quando il cliente che riceve il servizio da un server secondario è sempre soddisfatto. Il valore minimo si ottiene quando $q = 0$ e $\nu = 0$. Questa misura aumenta quando aumenta $q$ o $\nu$, e il tasso di aumento diventa più elevato quando uno o entrambi si avvicinano al valore 1. Quando $q = 1$, il sistema si trasforma nel corrispondente modello di coda MAP/M/1 classico e in un sistema senza l'uso del server secondario, e $L_\mathrm{system} = 22,30425$ per tutti i valori di $\nu$ (come è evidente). Quando $q = 0$, che corrisponde al caso in cui un cliente servito viene sempre reclutato (quando necessario), anche quando la probabilità di insoddisfazione è alta ($\nu = 0,5$), il valore di $L_\mathrm{system}$ è pari a $12,91247$. Pertanto, l'uso di un server secondario riduce essenzialmente il numero medio di clienti nel sistema di più del $40\%$. Inoltre, abbiamo cercato il punto di interruzione, diciamo $\nu^*$, per una percentuale di insoddisfazione tale per cui il modello classico di coda è migliore del modello proposto qui. Per i parametri di questo esempio, il punto di interruzione è $\nu^* \sim 0,985$, nel senso che la percentuale di insoddisfazione deve essere superiore al $98,5\%$ affinché il modello classico funzioni meglio. \vspace{0.4cm} + +\begin{figure}[h] + \centering + \includegraphics[width=0.6\textwidth]{0uFWo7C.png} + \caption{Dependence of the average number of customers with the secondary server $L_\mathrm{sec}$ on the parameters $q$ and $\nu$} + \label{fig:7} +\end{figure} + +Per testare ulteriormente l'importo della riduzione nel numero medio, abbiamo aumentato $\lambda$ del $50\%$ a $\lambda = 0.75$. Mantenendo tutti gli altri parametri (ad eccezione della normalizzazione dei parametri del processo di arrivo per ottenere questo specifico $\lambda$) gli stessi, abbiamo ottenuto una percentuale di riduzione superiore al $52,8\%$. Pertanto, un aumento del carico del sistema beneficerà notevolmente dell'avere un server secondario per aiutare il sistema anche con un tasso di insoddisfazione del cliente del $50\%$ con questo server secondario. +La figura \ref{fig:7} mostra la dipendenza del numero medio di clienti con il server secondario $L_\mathrm{sec}$ dai parametri $q$ e $\nu$. Questa probabilità diminuisce significativamente quando $q$ si avvicina a $1$ e quando i clienti sono raramente reclutati per diventare server secondari. $L_\mathrm{sec}$ ha il valore massimo quando $q$ è uguale a zero, ovvero tutti i clienti vengono reclutati (quando necessario) per diventare server secondari, e quando $\nu$ è vicino a 1. Ovviamente, in quest'ultimo caso, quasi tutti i clienti serviti da un server secondario devono essere rimandati al sistema a causa della loro insoddisfazione. Questo spiega la creazione di ulteriore lavoro per il sistema e dovrebbe essere scoraggiato ricorrendo alla coda classica anziché reclutare server secondari "scarsi". Vale la pena sottolineare che un sistema del genere (scadente) può riflettersi negativamente sul sistema stesso per la fornitura di servizi che non possono essere replicati da altri clienti serviti. + +\begin{figure}[h!] + \centering + \includegraphics[width=0.6\textwidth]{3zlg6uj.png} + \caption{Dependence of the probability $P_\mathrm{idle-system}$ that the system is idle at an arbitrary moment on the parameters $q$ and $\nu$.} + \label{fig:8} +\end{figure} + +Nella Figura \ref{fig:8}, è mostrato il comportamento della probabilità $P_\mathrm{idle-system}$ che il sistema è inattivo in un momento arbitrario come funzione di $q$ e $\nu$. Questa probabilità ha il valore minimo di $0.4445$ quando $\nu = 1$ e $q = 0$, il che è intuitivamente chiaro, poiché dover servire nuovamente i clienti dopo aver passato attraverso un server secondario mette un carico sul sistema. La probabilità che $P_\mathrm{idle-system}$ aumenta quando $q$ aumenta e/o $\nu$ diminuisce: il valore massimo $0.5652$ di questa probabilità si ottiene quando $q = 0.65$ e $\nu = 0$. Nel corrispondente sistema MAP/M/1 classico, questa misura è $P_\mathrm{idle-system} = 0.5$. + +\vspace{0.4cm} + +\noindent \textsc{Altre cose da dire su questo esempio numerico che secondo me si possono saltare. Tornare dopo} + +\clearpage +\subsection{Esempio numerico 3} + +In questo ultimo esempio, analizziamo l'impatto della variazione dei tassi di servizio $\mu_1$ e $\mu_2$ quando tutti gli altri parametri sono fissati. A tal fine, fissiamo $L = 10$, $q = 0.5$, $\nu = 0.4$ e $\lambda = 0.5$. I tassi $\mu_1$ e $\mu_2$ vengono variati da $0.25$ a $2.0$ con incrementi di $0.05$. È importante menzionare che, per soddisfare la condizione di ergodicità (vedi Equazione \ref{eq:ergodicity}), limitiamo ulteriormente il valore di $\mu_2$ quando $\mu_1$ è piccolo. In particolare, quando $\mu_1 = 0.25$, il valore minimo del tasso $\mu_2$ (con il passo sopra descritto di $0.05$) è scelto in modo tale da non essere inferiore a $0.65$. Quando $\mu_1 = 0.3$, il tasso $\mu_2$ è scelto in modo tale da non essere inferiore a $0.45$. Quando $\mu_1 = 0.35$, il tasso $\mu_2$ è scelto in modo tale da non essere inferiore a $0.3$. Solo per $\mu_1 \geq 0.4$, il valore di $\mu_2$ può essere variato da $0.25$, come originariamente indicato. + + +Con le sopra descritte restrizioni sulla scelta di $\mu_1$ e $\mu_2$, mostriamo nelle Figure \ref{fig:9} e \ref{fig:10} la dipendenza della misura $L_{\text{system}}$ da $\mu_1$ e $\mu_2$. Nella Figura \ref{fig:9}, gran parte della superficie che mostra la dipendenza appare piatta. Ciò è dovuto al fatto che, per molte combinazioni dei valori dei parametri con un piccolo tasso $\mu_1$, la condizione di ergodicità viene violata e la misura $L_{\text{system}}$ diventa molto grande. Pertanto, nella Figura \ref{fig:10}, la dipendenza di $L_{\text{system}}$ da $\mu_1$ e $\mu_2$ è mostrata solo per valori non piccoli di $\mu_1$. Chiaramente, si può notare una tendenza decrescente, poiché $L_{\text{system}}$ diminuisce rapidamente quando $\mu_1$ aumenta per $\mu_2$ fissato e viceversa. + +\begin{figure}[h] + \centering + \includegraphics[width=0.6\textwidth]{9LBRzWR.png} + \caption{Dependence of the average number of customers in the system $L_\mathrm{system}$ on the parameters + $\mu_1$ and $\mu_2$} + \label{fig:9} +\end{figure} + +\begin{figure}[h] + \centering + \includegraphics[width=0.6\textwidth]{CBOSr9C.png} + \caption{Dependence of the average number of customers in the system $L_\mathrm{system}$ on the parameters + $\mu_1$ and $\mu_2$.} + \label{fig:10} +\end{figure} + +La Figura \ref{fig:11} mostra il comportamento del numero medio di clienti con il server +secondario $L_\mathrm{sec}$. Il valore di $L_\mathrm{sec}$ è massimizzato con un valore di circa 5 quando $\mu_1$ e $\mu_2$ sono piccoli. Questo è intuitivamente chiaro poiché per valori piccoli di $\mu_1$ e $\mu_2$, la condizione di ergodicità è vicina a essere violata, causando un alto tasso di reclutamento per i server secondari che, molto probabilmente prima di lasciare il sistema, serviranno un gruppo di dimensione $L = 10$. Pertanto, il numero medio di clienti in servizio in un momento arbitrario è di circa 5. Con un aumento di $\mu_1$ e $\mu_2$, il valore di $L_\mathrm{sec}$ diminuisce come ci si aspetterebbe. Per valori piccoli di $\mu_1$, la diminuzione è significativa all'aumentare di $\mu_2$; per valori più grandi di $\mu_1$, notiamo un tasso insignificante di diminuzione in $L_\mathrm{sec}$ con un aumento di $\mu_2$. + +\begin{figure}[h] + \centering + \includegraphics[width=0.6\textwidth]{R1VGHAR.png} + \caption{Dependence of the average number of customers with the secondary server L sec on the + parameters $\mu_1$ and $\mu_2$.} + \label{fig:11} +\end{figure} + +\vspace{0.4cm} + +\noindent \textsc{Altre cose da dire su questo esempio numerico che secondo me si possono saltare. Tornare dopo} + +\clearpage +\section{Conclusioni} +In questo articolo, è stato analizzato un sistema di coda in cui c'è la possibilità di reclutare un cliente già servito come server secondario per aiutare il server principale assegnando un gruppo di clienti in attesa. Il processo di arrivo è stato modellizzato utilizzando un processo di punto Markoviano versatile, MAP. È stata presa in considerazione la possibilità di insoddisfazione dei clienti con il servizio fornito dal server secondario, causando il ritorno di quei clienti nel sistema. È stata implementata l'analisi dello stato stazionario della catena di Markov multidimensionale che descrive il comportamento del sistema e sono stati presentati risultati numerici illustrativi potenzialmente utili per prendere decisioni manageriali. +Il modello studiato in questo articolo può essere generalizzato in diversi modi. Ad esempio, +\begin{enumerate} + \item il servizio fornito dal server secondario può essere effettuato in gruppi; + \item rilassare l'ipotesi di avere solo un server secondario a più di uno e vedere l'impatto dell'aumento a, diciamo, 2; + \item utilizzare servizi di tipo fase-possibilmente con rappresentazioni diverse per il server principale e secondario; + \item incorporare l'impazienza dei clienti sia nei buffer principali che secondari; + \item implementare un processo di reclutamento in base alla lunghezza della coda osservata basato su una politica di controllo di tipo soglia; + \item consentire arrivi di gruppo; e infine + \item incorporare la possibilità di reclutare molti server secondari con due tipi di clienti in modo che solo un tipo possa agire come server secondario. +\end{enumerate} + +\end{document} diff --git a/discorso/pSUZiNk.png b/discorso/pSUZiNk.png new file mode 100644 index 0000000000000000000000000000000000000000..d50e7caab421bb92d003641d0db113170593a529 GIT 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z-nv@8zeh~rMtNB4S4gd5zo|kljAl*VZngryHxo2~|JF^KOzaRJ7-6cTxe+lL;guMX zS^m1NL6-XuYX?B9`4ShsTj5+l8=>O#Od2{0YQnd58B~C3eL96jnbS5$foQ(`(gRv2 zM3=}10v<~|O8^2IgK~2K@$3!Gm5PK%C;IQgC;=ZR1XB4?7TttK@T=rnZDLtlw&u8^ ze-S^Lb9wtQpb6{a*&RZow6wK9$MD~T?>~={IJMq{+XO@PH_{9@ zwIx40eHo7L{;(@Kdiyq=3Bz_8FSg~)=NAt?U<|kldFx-w8%p(vM1w=Io4Po z#XA=$oEoW-39o`>5yubDMALEd6zI2sGPo}LW9AaL<^0AHtNl`en&5AIr!w-a<1M;O z9Qp(yp%SpWuk{87X?6Cx;1b;7br$ehygk+LEKtSc`)u{&#sx`tIo5NF|183)OP)NC znJF$LTN=q&wz2eJXFYcy4VWBD?>G+54@x9*ZZT76gC2zHORdGW+EXX%g9BFx&K}v2 zOW%#(e;%~|lM2{whr5AejX6kb`hvvqi!n;WAaUYZbfz%9ocJo?OD1+2_t>-hr>a5q gKj9W}1nN5g-#UOF*fKG4a y Q^+e + \end{equation*} + dove il vettore $y$ è l'unica soluzione del sistema + \begin{equation*} + y(Q^- + Q^0 + Q^+) = \textbf{0}, \qquad ye = 1 + \end{equation*} + + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{} + \begin{block}{Dimostrazione} + Si può inoltre verificare facilmente che + \begin{equation*}\label{eq:bigotimes} + Q^- + Q^0 + Q^+ = I_{L+1} \otimes (D_0 + D_1) + S \otimes I_{m} + \end{equation*} + dove + \begin{equation*} + S = + \begin{pmatrix} + -\mu_1 (1-q) & 0 & 0 & \dots & 0 \mu_1(1-q) \\ + \mu_2 & -\mu_2 & 0 & \dots & 0 & 0 \\ + 0 & \mu_2 & -\mu_2 & \dots & 0 & 0 \\ + \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ + 0 & 0 & 0 & \dots & \mu_2 & -\mu_2 \\ + \end{pmatrix} + \end{equation*} + \end{block} +\end{frame} + +\begin{frame} + \frametitle{} + \begin{block}{Dimostrazione} + dove usando le regole del mixed product per il prodotto di Kronecker, e ricordando che + $$ \delta(D_0 + D_1) = 0, \qquad \delta e = 1 $$ + si verifica facilmente caratteristiche + $$ y = x \otimes \delta $$ + dove $x$ è soluzione del sistema + $$xS = 0, \qquad xe = 1 $$ + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{} + \begin{block}{Dimostrazione} + per sostituzione diretta, si verifica facilmente che le componenti del vettore $x = (x_0, x_1, ... , x_L)$, corrispondenti alle uniche soluzioni del sistema visto prima, sono date da + \begin{equation*} + x_0 = \frac{\mu_2}{L(1-q)\mu_1 + \mu_2}, \qquad x_i = \frac{\mu_1(1-q)}{L(1-q)\mu_1 + \mu_2}, \qquad i = 1, ..., L + \end{equation*} + La tesi segue delle equazioni viste in precedenza assieme alla definizione di $\lambda$. + \qed + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{Osservazioni sulla dimostrazione} + \begin{block}{Osservazione 1} + \begin{itemize} + \item La condizione di ergodicità richiede che il tasso di arrivo dei clienti per unità di tempo debba essere inferiore al tasso di servizio che i clienti ricevono per unità di tempo quando il sistema è sovraccarico. + \item Il tasso di servizio medio totale nel modello di coda è dato dalla somma del tasso di servizio fornito dal server principale e del tasso di servizio fornito dal server secondario. + \end{itemize} + Possiamo esprimere il tasso di servizio medio totale come segue: + $$\mu = \mu_1 + \mu_2(1 - \nu) \frac{L(1-q)\mu_1}{L(1-q)\mu_1 + \mu_2}$$ + \end{block} +\end{frame} + +\begin{frame} + \frametitle{Osservazioni sulla dimostrazione} + \begin{block}{Osservazione 2} + Calcoliamo la probabilità $x_0$ che il secondo server non sia presente nel sistema in un qualsiasi momento in cui il sistema è sovraccarico. + + \begin{itemize} + \item Quando il sistema attiva un server secondario la durata media del server secondario continuamente presente nel sistema è data da $\frac{L}{\mu_2}$. Pertanto, abbiamo: + \begin{equation*} + x_0 = \frac{\frac{1}{\mu_1(1-q)}}{\frac{1}{\mu_1(1-q)} + \frac{L}{\mu_2}} = \frac{\mu_2}{L(1-q)\mu_1 + \mu_2} + \end{equation*} + \end{itemize} + \end{block} +\end{frame} + +\subsubsection{Calcolo della distribuzione stazionaria del processo QDB} + +\begin{frame} + \frametitle{Probabilità stazionarie} + Sotto l'assunzione che la condizione di ergodicità sia valida, esistono le seguenti probabilità stazionarie degli stati del CTMC $\{\zeta_t, t \geq 0\}$: + $$\pi(i,n,\zeta) = \lim_{t \to \infty} P\{i_t = i, n_t = n, \zeta_t = \zeta\}, ~ i \geq 0$$ + % $$n \in \{0, 1, \dots , \min\{i,L\}\}, \quad \zeta \in \{0, \dots, n\}$$ + Consideriamo i vettori riga delle probabilità di stato stazionario $\pi_i$ come segue + \begin{block}{} + $$\pi(i,n) = (\pi(i,n,1), ..., \pi(i,n,m))$$ + $$\pi_i = (\pi(i,0), ..., \pi(i, \min \{i,L\})), \quad i \geq 0$$ + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{Probabilità stazionarie} + Sappiamo che i vettori di probabilità stazionari $\pi_i, i \geq 0$, soddisfano il sistema di equazioni algebriche lineari (equazioni di equilibrio): + $$(\pi_0, \pi_1, \pi_, \dots)Q = 0 \qquad (\pi_0, \pi_1, \pi_, \dots)e = 1$$ + dove $Q$ è la matrice di transizione del CTMC $\{\zeta_t, t \geq 0\}$ e $e$ è il vettore colonna di tutti gli elementi $1$ +\end{frame} + +% La soluzione del problema di calcolo della distribuzione stazionaria di una QBD indipendente dal livello è ben nota; si veda [37]. Per i livelli in cui le transizioni della QBD non dipendono dal livello, i vettori di probabilità stazionaria sono trovati in forma matriciale geometrica. I vettori di probabilità stazionaria dei livelli di confine, in cui le transizioni della QBD dipendono dal livello, sono quindi direttamente trovati come soluzione del sistema di equazioni algebriche lineari. Tuttavia, se il numero di livelli di confine è grande (cosa che accade nel nostro modello se $L$ è grande), questo sistema ha una grande dimensione + +\begin{frame} + \frametitle{Algoritmo per risolvere il sistema di equazioni di equilibrio} + \begin{block}{} + Vediamo un algoritmo per risolvere il sistema infinito di equazioni di equilibrio che sfrutta la struttura tridiagonale a blocchi ma dipendente dal livello del generatore per i livelli minori di $L + 1$. + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{Algoritmo per risolvere il sistema di equazioni di equilibrio} + \begin{theorem} + I vettori $\pi_i, i \geq 0$, sono trovati come soluzione del sistema di equazioni algebriche lineari: + \begin{equation*} + \pi_i = \alpha_i \big( \sum_{l=0}^\infty \alpha_l e \big)^{-1}, \qquad i \geq 0 + \end{equation*} + dove il vettore $\alpha_0$ è calcolato come l'unica soluzione del sistema di equazioni + \begin{equation*} + \alpha_0(Q_{0,0} + Q_{0,1}G_0) = 0, \qquad \alpha_0e = 1 + \end{equation*} + ed i vettori $\alpha_i, i \geq 1$, sono definiti come + \begin{equation*} + \alpha_i = \alpha_0 \prod_{l=1}^i R_l, \qquad i \geq 1 + \end{equation*} + \end{theorem} +\end{frame} + + +\begin{frame} + \frametitle{Algoritmo per risolvere il sistema di equazioni di equilibrio} + \begin{theorem} + Altrimenti tramite la formula ricorsiva + \begin{equation*}\label{eq:rec-alpha} + \alpha_i = \alpha_{i-1} R_i, \qquad i \geq 1 + \end{equation*} + dove + \begin{equation*} + R = + \begin{cases} + -Q_{i-1,i}(Q_{i,i} + Q_{i,i+1}G_i)^{-1}Q & 1 \leq i \leq L-1 \\ + -Q_{L-1,L}(Q_{L,L} + Q^+G)^{-1} & i = L \\ + -Q^+(Q^0 + Q^+G)^{-1} = R & i>L + \end{cases} + \end{equation*} + \end{theorem} +\end{frame} + + +\begin{frame} + \frametitle{Algoritmo per risolvere il sistema di equazioni di equilibrio} + \begin{theorem} + Le matrici stocastiche $G_i$ sono calcolate utilizzando la seguente formula ricorsiva all'indietro: + $$G_L = G$$ + $$G_L-1 = -(Q_{L,L} + Q^+G_L)^{-1}Q_{L,L-1}$$ + $$G_i = -(Q_{i+1,i+1} + Q_{i+1,i+2}G_{i+1})^{-1}Q_{i+1,i}, \qquad i = L-2, L-3, \dots, 0$$ + dove la matrice $G$ è la minima soluzione non negativa dell'equazione quadratica matriciale + \begin{equation*} + Q^+G^2 + Q^0G + Q^- = 0 + \end{equation*} + \end{theorem} +\end{frame} + + +\begin{frame} + \frametitle{Algoritmo per risolvere il sistema di equazioni di equilibrio} + \begin{itemize} + \item L'algoritmo proposto è una modifica dell'algoritmo per il calcolo della distribuzione stazionaria del CTMC asintoticamente quasi-Toeplitz. + \item Utilizzando la ricorsione di vettori anziché quella di matrici si ha una significativa riduzione della memoria del computer e del tempo di esecuzione. + \item Le inverse delle matrici utilizzate nell'algoritmo sono sub-generatori irriducibili e semi-stabili, il che rende stabile l'implementazione numerica dell'algoritmo. + \end{itemize} +\end{frame} + + +\subsection{Approccio G1/M/1} + + +\begin{frame} + \frametitle{Code di tipo GI/M/1} + \begin{block}{} + Definiamo come prima cosa lo spazio degli stati $\Omega$ del CTMC come: + $$\Omega = \{ (i,j,k) ~ : ~ i \geq 0, 0 \leq j \leq K, 1 \leq k \leq m \}$$ + \end{block} + \begin{block}{} + Definiamo il livello + $$\textbf{i} = \{(i, j, k) : 0 \leq j \leq L, 1 \leq k \leq m\} = \{(\textbf{i}, 0), \dots, (\textbf{i}, L)\}, i \geq 0$$ + \begin{itemize} + \item il livello $(\textbf{i},\textbf{j})$ indica che il server principale è occupato, ci sono $i-1$ clienti in attesa nella coda principale; il server secondario è occupato e il processo di arrivo si trova in varie fasi + \item Il livello $(\textbf{0},\textbf{0})$ corrisponde al sistema inattivo con il processo MAP in una delle $m$ fasi. + \end{itemize} + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{Il generatore del CTMC} + Il generatore $\widetilde{Q}$ della CTMC che governa il sistema in studio è: + \begin{figure} + \centering + \includegraphics[width=\textwidth]{Isy9B7s.png} + \end{figure} +\end{frame} + + +\begin{frame} + \frametitle{Il generatore del CTMC} + Dove abbiamo: + \begin{equation*} + B_0 = + \begin{pmatrix} + D_0 & & & & \\ + \widetilde{\nu}\mu_2 I & D_0 - \mu_2 I & & & \\ + & \widetilde{\nu}\mu_2 I & D_0 - \mu_2 I & & \\ + & & \ddots & \ddots & \\ + & & & \widetilde{\nu}\mu_2 I & D_0 - \mu_2 I + \end{pmatrix} + \end{equation*} +\end{frame} + + +\begin{frame} + \frametitle{Il generatore del CTMC} + Dove abbiamo: + \begin{equation*} + \begin{split} + A_0 &= + \begin{pmatrix} + D_1 & & & & \\ + \nu\mu_2 I & D_1 & & & \\ + & \nu \mu_2 I & D_1 & & \\ + & & \ddots & \ddots & \\ + & & & \nu \mu_2 I & D_1 + \end{pmatrix} \\ + A_1 &= B_0 -\mu_1 I \\ + A_2 & = \mu_1 \Delta(q,1, \dots, 1) \\ + B_1 &= \mu_1 I \\ + B_r &= \rho \mu_1 (e_r^T \otimes e(L+1)) \qquad 2 \leq r \leq L+1 \\ + A_{L+2} &= B_{L+1} + \end{split} + \end{equation*} +\end{frame} + + +\begin{frame} + \frametitle{Proprietà delle queue di tipo GI/M/1} + \begin{block}{Proprietà 1} + Sia + $$\widetilde{y} = (\widetilde{y_0}, \dots, \widetilde{y_{L}})$$ + il vettore invariante di $\displaystyle A = \sum_{i=0}^{L+2} A_i$. Allora: + $$ \widetilde{y_0} = \delta(\mu_2I - D_0 - D_1)[\mu_2U + L\rho\mu_1 I - D_0 - D_1]^{-1} $$ + $$ \widetilde{y_r} = \rho \mu_1 \pi_0(\mu_2I - D_0 - D_1)^{-1}, \qquad 1 \leq r \leq L $$ + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{Proprietà delle queue di tipo GI/M/1} + \begin{block}{Proprietà 2} + La condizione di stabilità + $$ \widetilde{y} A_0 e < \widetilde{y} \sum_{i=1}^{L+2}(i-1)A_i e $$ + si riduce a: + $$ \lambda < \mu_1 + \mu_2(1 - \nu) \frac{L(1-q)\mu_1}{L(1-q)\mu_1 + \mu_2} $$ + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{Proprietà delle queue di tipo GI/M/1} + \begin{block}{Proprietà 3} + Data $R$ la matrice di rate, soddisfa l'equazione matriciale non lineare data da: + $$ R^{L+2} A_{L+2} + R^2A_2 + RA_1 + A_0 = 0 $$ + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{Proprietà delle queue di tipo GI/M/1} + \begin{block}{Proprietà 4} + Indicando con $\widetilde{\pi}$ il vettore di probabilità in stato stazionario del generatore $\widetilde{Q}$ come visto prima, otteniamo qui la soluzione matrice-geometrica classica: + $$ \widetilde{\pi}_i = \widetilde{\pi}_0R^i, \qquad i \geq 1 $$ + dove $\widetilde{\pi}_0$ è ottenuto risolvendo il seguente sistema di equazioni lineari: + $$ \widetilde{\pi}_0\Bigg[ \sum_{i=0}^{L+1} R^iB_i \Bigg] = 0, \qquad \widetilde{\pi}_0e = 1 $$ + \end{block} +\end{frame} + + +\section{Risultati Numerici} +\begin{frame} + \frametitle{Introduzione ai risultati numerici} + Vedremo 3 elementi illustrativi utilizzando 5 processi di arrivo. In particolare prendiamo i 5 MAP come + \begin{block}{ERL} + Erlang di ordine 5 con parametro 2.5 in ciascuno dei 5 stati. Notare che qui abbiamo $\lambda = 0.5, \sigma = 0.899427$ e $\rho_c = 0$. + \end{block} + \begin{block}{EXP} + Un esponenziale con una frequenza di 0.5. Notare che qui abbiamo $\lambda = 0.5, \sigma = 2$ e $\rho_c = 0$. + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{} + \begin{block}{HEX} + Distribuzione iper-esponenziale con una probabilità di mixing data da (0.5, 0.3, 0.15, 0.04, 0.01) con i corrispondenti tassi della distribuzione esponenziale pari a (1.09, 0.545, 0.2725, 0.13625, 0.068125). Qui abbiamo $\lambda = 0.5, \sigma = 3.3942$ e $\rho_c = 0$. + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{} + \begin{block}{NCR} + MAP negativamente correlato, con matrici di rappresentazione: + \begin{equation*} + D_0 = + \begin{pmatrix} + -1.125 & 0.125 & 0 & 0 & 0 \\ + 0 & -1.125 & 0.125 & 0 & 0 \\ + 0 & 0 & -1.125 & 0.125 & 0 \\ + 0 & 0 & 0 & -0.125 & 0 \\ + 0 & 0 & 0 & 0 & -2.25 + \end{pmatrix} + \end{equation*} + \begin{equation*} + D_1 = + \begin{pmatrix} + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0.01125 & 0 & 0 & 0 & 1.11375 \\ + 2.2275 & 0 & 0 & 0 & 0.0225 + \end{pmatrix} + \end{equation*} + dove abbiamo $\lambda = 0.5, \sigma = 2.02454$ e $\rho_c = -0.57855$ + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{} + \begin{block}{PCR} + MAP positivamente correlato, con matrici di rappresentazione: + \begin{equation*} + D_0 = + \begin{pmatrix} + -1.125 & 0.125 & 0 & 0 & 0 \\ + 0 & -1.125 & 0.125 & 0 & 0 \\ + 0 & 0 & -1.125 & 0.125 & 0 \\ + 0 & 0 & 0 & -0.125 & 0 \\ + 0 & 0 & 0 & 0 & -2.25 + \end{pmatrix} + \end{equation*} + \begin{equation*} + D_1 = + \begin{pmatrix} + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 1.11375 & 0 & 0 & 0 & 0.01125 \\ + 0.0225 & 0 & 0 & 0 & 2.2275 + \end{pmatrix} + \end{equation*} + dove abbiamo $\lambda = 0.5, \sigma = 2.02454$ e $\rho_c = -0.57855$ + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{Introduzione ai risultati numerici} + \begin{block}{Osservazioni} + \begin{itemize} + \item Le cinque MAP sopra riportate sono qualitativamente diverse. + \item Il processo di arrivo denominato PCR è ideale per situazioni di arrivi altamente irregolari. + \item La correlazione positiva nel processo PCR ha un impatto significativo e la variabilità nei tempi tra gli arrivi è stata ben documentata in letteratura. + \end{itemize} + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{Primo esempio illustrativo} + Fissiamo $\mu_1=1$, $\mu_2=0.5$, $q=0.5$, e $\nu=0.4$, e variamo $L$ da 1 a 30. + \begin{figure}[h] + \centering + \includegraphics[width=0.7\textwidth]{gtHnIzQ.png} + \caption{Impatto di $L$ sul numero medio di clienti nel sistema $L_{system}$ per diversi MAPs} + \label{fig:1} + \end{figure} +\end{frame} + + +\begin{frame} + \frametitle{Primo esempio illustrativo} + \begin{block}{$L_{\mathrm{sec}}$} + Definiamo $L_{\mathrm{sec}}$ come il numero medio di clienti nel sistema con server secondari ad un momento arbitrario come: + $$ L_{\mathrm{sec}} = \sum_{i=1}^{\infty} \sum_{n=1}^{\min\{i,L\}} n \pi (i,n) e $$ + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{Primo esempio illustrativo} + \begin{itemize} + \item La figura mostra che PCR ha un alto numero medio di clienti nel sistema rispetto ad altri processi di arrivo. + \item $L$ aumenta il numero medio di clienti nel sistema per i primi quattro MAP, ma per PCR il trend non è crescente a causa della correlazione positiva. + \item L'alta $L$ aumenta la probabilità di avere più clienti nel sistema, soprattutto per i primi quattro MAP. + \item Tuttavia, per gli arrivi PCR, $L$ diminuisce il numero medio di clienti nel sistema perché i server secondari aiutano a ripulire la coda. + \end{itemize} +\end{frame} + + +\begin{frame} + \frametitle{Primo esempio illustrativo} + \begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{Tcjl81t.png} + \caption{Dipendenza del numero medio di clienti con il server secondario $L_{\mathrm{sec}}$ al variare di $L$ per diversi MAPs} + \label{fig:2} + \end{figure} +\end{frame} + + +\begin{frame} + \frametitle{Primo esempio illustrativo} + \begin{itemize} + \item $L_{\mathrm{sec}}$ aumenta all'aumentare di L, come previsto. + \item Il valore di $L_{\mathrm{sec}}$ è elevato per PCR e solo per valori piccoli di L è inferiore per ERL-NCR. + \item L'alta irregolarità degli arrivi nel processo PCR causa la fame del sistema, mentre una maggiore varianza implica un grande numero di clienti nel sistema per ERL-HEX. + \end{itemize} +\end{frame} + + +\begin{frame} + \frametitle{Primo esempio illustrativo} + \begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{pSUZiNk.png} + \caption{Dipendenza della probabilità $P_{\mathrm{idle-system}}$ rispetto ad $L$ che il sistema sia in idle ad un momento arbitrario, per diversi MAPs} + \label{fig:3} + \end{figure} +\end{frame} + + +\begin{frame} + \frametitle{Primo esempio illustrativo} + \begin{block}{$P_{\mathrm{idle-system}}$} + Definiamo la probabilità che il sistema sia in equilibrio ad un momento arbitrario come: + $$ P_{\mathrm{idle-system}} = \pi_0 e $$ + \end{block} + \begin{itemize} + \item Esiste una grande differenza nella misura a seconda dei diversi MAPs utilizzati. + \item Il valore ottimale di $L$ dipende dall'obiettivo: ad esempio, per il processo di arrivo PCR, il valore ottimale di $L$ è $16$ se si cerca di minimizzare $L_{\mathrm{system}}$, ma è 6 se si massimizza $P_{\mathrm{idle-system}}$. + \end{itemize} +\end{frame} + + +\begin{frame} + \frametitle{Primo esempio illustrativo} + \begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{VrxLKYI.png} + \caption{Dipendenza della probabilità $P_{\mathrm{idle-busy}}$ rispetto ad $L$ che il main server sia in idle quando il server secondario è in occupato, per diversi MAPs} + \label{fig:4} + \end{figure} +\end{frame} + + +\begin{frame} + \frametitle{Primo esempio illustrativo} + \begin{figure} + \centering + \includegraphics[width=0.8\textwidth]{EZp90oP.png} + \caption{Dipendenza della probabilità $P_{\mathrm{busy-idle}}$ rispetto ad $L$ che il main server sia occupato quando il server secondario è in idle, per diversi MAPs} + \label{fig:5} + \end{figure} +\end{frame} + + +\begin{frame} + \frametitle{Primo esempio illustrativo} + \begin{block}{$P_{\mathrm{idle-busy}}$} + Definiamo la probabilità che il main server sia in idle quando il server secondario è occupato come: + $$ P_{\mathrm{idle-busy}} = \sum_{n=1}^{L} \pi(n,n) e $$ + \end{block} + \begin{block}{$P_{\mathrm{busy-idle}}$} + Definiamo la probabilità che il main server sia occupato quando il server secondario è in idle come: + $$ P_{\mathrm{busy-idle}} = \sum_{i=0}^{\infty} \pi(i,0) e $$ + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{Secondo esempio illustrativo} + \begin{itemize} + \item L'obiettivo è valutare l'impatto dei parametri $q$ e $\nu$ sulla prestazione del sistema. + \item Fissiamo il valore di $L$ a $10$ e i tassi di servizio $\mu_1$ e $\mu_2$ a $1$ e $0.5$. + \item Si variano i valori di $q$ e $\nu$ da $0$ a $1$ con passo $0.05$ e si analizza l'impatto sulle misure di prestazione del sistema. + \end{itemize} + \begin{block}{} + In questo esempio ci concentriamo sul processo di arrivo PCR, la cui scelta è basata sul comportamento di questo processo sulle misure evidenziato nel primo esempio illustrativo + \end{block} +\end{frame} + + +\begin{frame} + \frametitle{Secondo esempio illustrativo} + \begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{qUQVrEZ.png} + \caption{Dipendenza del numero medio di clienti nel sistema $L_{\mathrm{system}}$ rispetto a $q$ e $\nu$} + \end{figure} +\end{frame} + + +\begin{frame} + \frametitle{Secondo esempio illustrativo} + \begin{itemize} + \item L'analisi mostra che il valore di $L_\mathrm{system}$ è minimo a $7.9328$ quando $q$ e $\nu$ sono entrambi uguali a $0$. + \item Aumentando $q$ o $\nu$, il valore di $L_\mathrm{system}$ aumenta, con un aumento più veloce quando uno o entrambi si avvicinano a $1$. + \item Quando $q=1$, il sistema diventa un modello MAP/M/1 classico e il valore di $L_\mathrm{system}$ diventa $22.30425$ per tutti i valori di $\nu$. + \item L'uso di un server secondario riduce il numero medio di clienti nel sistema di oltre il $40\%,$ e il punto di interruzione per il modello classico è $\nu^* \sim 0.985$. + \end{itemize} +\end{frame} + + +\begin{frame} + \frametitle{Secondo esempio illustrativo} + \begin{itemize} + \item Si aumenta $\lambda$ del 50\% a 0.75 per testare l'importo della riduzione del numero medio di clienti nel sistema. + \item Mantenendo gli altri parametri costanti, si ottiene una riduzione superiore al 52,8\%. + \item Ciò suggerisce che l'aggiunta di un server secondario beneficia notevolmente l'aumento del carico del sistema anche con un tasso di insoddisfazione del cliente del 50\%. + \end{itemize} +\end{frame} + + +\begin{frame} + \frametitle{Secondo esempio illustrativo} + \begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{0uFWo7C.png} + \caption{ Dipendenza del numero medio di clienti nel sistema $L_{\mathrm{sec}}$ rispetto a $q$ e $\nu$ con $\lambda=0.75$} + \end{figure} +\end{frame} + + +\begin{frame} + \frametitle{Secondo esempio illustrativo} + \begin{itemize} + \item La figura mostra come il numero medio di clienti con il server secondario $L_\mathrm{sec}$ dipenda dai parametri $q$ e $\nu$. + \item $L_\mathrm{sec}$ diminuisce significativamente quando $q$ si avvicina a $1$ e quando i clienti sono raramente reclutati per diventare server secondari. + \item $L_\mathrm{sec}$ ha il valore massimo quando $q$ è uguale a zero e $\nu$ è vicino a $1$, ma questo può creare ulteriore lavoro per il sistema e riflettersi negativamente sulla fornitura di servizi. + \end{itemize} +\end{frame} + + +\begin{frame} + \frametitle{Secondo esempio illustrativo} + \begin{figure}[h!] + \centering + \includegraphics[width=0.8\textwidth]{3zlg6uj.png} + \caption{Dipendenza della probabilità $P_\mathrm{idle-system}$ che il sistema sia in idle ad un momento arbitrario rispetto a $q$ e $\nu$.} + \end{figure} +\end{frame} + + +\begin{frame} + \frametitle{Secondo esempio illustrativo} + \begin{itemize} + \item $P_\mathrm{idle-system}$ ha il valore minimo di $0.4445$ quando $\nu=1$ e $q=0$, il che è intuitivo poiché servire nuovamente i clienti dopo aver passato attraverso un server secondario mette un carico sul sistema. + \item $P_\mathrm{idle-system}$ aumenta quando $q$ aumenta e/o $\nu$ diminuisce, con un valore massimo di $0.5652$ ottenuto quando $q=0.65$ e $\nu=0$. + \item Nel sistema MAP/M/1 classico corrispondente, questa misura è $P_\mathrm{idle-system}=0.5$. + \end{itemize} +\end{frame} + + +\begin{frame} + \frametitle{Terzo esempio illustrativo} + \begin{itemize} + \item In questo esempio, si analizza l'impatto della variazione dei tassi di servizio $\mu_1$ e $\mu_2$ quando tutti gli altri parametri sono fissati. + \item I parametri fissati sono $L=10$, $q=0.5$, $\nu=0.4$, e $\lambda=0.5$. + \item I tassi $\mu_1$ e $\mu_2$ vengono variati da $0.25$ a $2.0$ con incrementi di $0.05$, ma per soddisfare la condizione di ergodicità, il valore di $\mu_2$ viene limitato quando $\mu_1$ è piccolo. + \item Solo per $\mu_1\geq0.4$, il valore di $\mu_2$ può essere variato da $0.25$, come originariamente indicato + \end{itemize} +\end{frame} + + +\begin{frame} + \frametitle{Terzo esempio illustrativo} + \begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{9LBRzWR.png} + \caption{Dipendenza del numero medio di clienti nel sistema $L_{\mathrm{system}}$ rispetto a $\mu_1$ e $\mu_2$} + \end{figure} +\end{frame} + + +\begin{frame} + \frametitle{Terzo esempio illustrativo} + \begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{CBOSr9C.png} + \caption{Dipendenza del numero medio di clienti nel sistema $L_{\mathrm{system}}$ rispetto a $\mu_1$ e $\mu_2$} + \end{figure} +\end{frame} + + +\begin{frame} + \frametitle{Terzo esempio illustrativo} + \begin{itemize} + \item La condizione di ergodicità limita il valore di $\mu_2$ per valori piccoli di $\mu_1$. + \item Gran parte della superficie mostrata nella prima figura è piatta a causa della violazione della condizione di ergodicità. + \item La tendenza generale è che $L_{\text{system}}$ diminuisce quando aumenta $\mu_1$ o $\mu_2$ (per valori non piccoli di $\mu_1$). + \end{itemize} +\end{frame} + + +\begin{frame} + \frametitle{Terzo esempio illustrativo} + \begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{R1VGHAR.png} + \caption{Dipendenza del numero medio di clienti con server secondario $L_{\mathrm{sec}}$ rispetto a $\mu_1$ e $\mu_2$} + \end{figure} +\end{frame} + + +\begin{frame} + \frametitle{Terzo esempio illustrativo} + + \begin{itemize} + \item Il valore massimo di $L_{sec}$ è di circa 5 quando $\mu_1$ e $\mu_2$ sono piccoli. + \item Con un aumento di $\mu_1$ e $\mu_2$, il valore di $L_{sec}$ diminuisce come ci si aspetterebbe. + \item Per valori piccoli di $\mu_1$, la diminuzione è significativa all'aumentare di $\mu_2$; per valori più grandi di $\mu_1$, notiamo un tasso insignificante di diminuzione in $L_{sec}$ con un aumento di $\mu_2$. + \end{itemize} + + +\end{frame} + + +\section{Conclusioni} + +\begin{frame} + \frametitle{Conclusioni} + \begin{itemize} + \item Il sistema di coda analizzato prevede la possibilità di reclutare un cliente già servito come server secondario per aiutare il server principale. + \item Il processo di arrivo dei clienti è stato modellizzato utilizzando un processo di punto Markoviano versatile, MAP. + \item È stata considerata la possibilità che i clienti insoddisfatti con il servizio fornito dal server secondario possano ritornare nel sistema. + \item L'analisi dello stato stazionario della catena di Markov multidimensionale ha permesso di ottenere risultati numerici utili per prendere decisioni manageriali. + \end{itemize} +\end{frame} + + +\begin{frame} + \frametitle{Generalizzazione del modello} + \begin{itemize} + \item Si può effettuare il servizio fornito dal server secondario in gruppi. + \item Si può rilassare l'ipotesi di avere solo un server secondario e vedere l'impatto dell'aumento a 2. + \item Si possono incorporare diverse politiche di controllo, come la possibilità di reclutare molti server secondari con due tipi di clienti + \end{itemize} +\end{frame} + + + + + +\end{document} diff --git a/slide/pSUZiNk.png 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