Optimal control of an M/M/2 queueing system with finite capacity operating under the triadic (0, Q, N, M) policy

In this paper, we study two removable servers in the controllable M/M/2 queueing system with finite capacity L operating under the triadic (0, Q, N, M) policy, where L is the maximum number of customers in the system. The number of working servers can be adjusted one at a time at arrival epochs or at service completion epochs depending on the number of customers in the system. Analytic closed-form solutions of the controllable M/M/2 queueing system with finite capacity operating under the triadic (0, Q, N, M) policy are derived. This is a generalization of the ordinary M/M/2 and the controllable M/M/1 queueing systems in the literature. The total expected cost function per unit time is developed to obtain the optimal operating (0, Q, N, M) policy at minimum cost

optimal control of an M/M/2 queueing system with finite capacity operating under the triadic (0,Q,N,M) policy

本篇論文討論有限容量為L的M/M/2排隊系統含有兩個可移動服務站之最佳控制，其中控制決策為(0, Q, N, M)-方策。我們假設顧客到達率服從卜瓦松過程;而服務站服務顧客的時間則呈現負指數分佈。所謂(0, Q, N, M)-方策，是指當系統中顧客數目累積到N時，一個服務站立即被開啟並提供服務，之後，當系統中的顧客數目繼續增加至M，則第二個服務站也立即被開啟並提供服務。然而，當兩個服務站皆正在運作當中，而系統中的顧客數目卻減少至Q時，即立刻關閉一個服務站。此時只剩下一個服務站正在提供服務，若此時的顧客數目繼續減少至0，則服務站立即被完全關閉。直到上面所述的情況發生，服務站才會重新被開啟並提供服務。我們首先求出其穩態解並計算其系統特徵值。其次，證明(0, Q, N, M) -方策之下的M/M/2排隊系統會比下列兩個排隊系統更一般化：(i)標準M/M/2排隊系統; (ii) N方策M/M/1排隊系統。最後，我們建立一個單位時間的總成本函數，然後決定出控制參數(Q, N, M)的最佳解 ，使成本函數達到最小值。In this thesis, we study the controllable M/M/2 queueing system with finite capacity L operating under the triadic (0, Q, N, M) policy, where L is the maximum capacity of the system.Customers arrive following a Poisson process and the service times are assumed to have the negative exponential distribution. The triadic (0, Q, N, M) policy means that we turn one server on when the number of customers waiting for service reaches N. At a later time, when the number of customers waiting for service increases to M(N<M), then the other server will also active instantly. However, if the number of customers in the system decreases to Q while both servers are active simultaneously, the server just finishing service is removed from the system at that time. Furthermore, if the number of customers in the system reaches to zero while one server is active, that server is turned off until the above conditions are occurred again. We first develop the steady-state analytic solutions in neat-closed form expressions and calculate the steady-state characteristics of the system. Next, we show that the controllable M/M/2 queueing system with finite capacity L operating under the triadic (0, Q, N, M) policy generalizes (i) the ordinary M/M/2 queueing system; and (ii) the controllable M/M/1 queueing system operating under the N policy. Finally, we construct the total expected cost function per unit time to determine the optimal value of the decision variables (Q, N, M) at minimum cost.1.Introduction ................................................1 1.1.Problem statement..........................................2 1.2.Literature review .........................................3 1.3.The scope of the study ....................................4 2.Steady-State Results ........................................5 1.4.Definition of the operating policy ........................5 1.5.Assumptions of the system .................................6 1.6.Steady-state solutions ....................................6 1.7.Probability generating functions ..........................9 1.8.Computation for , , and ...............................12 1.9.System characteristics ...................................13 1.10.Special cases of the system .............................15 3.Optimal Triadic (0,Q,N,M) Policy ...........................18 3.1.Computations for E[I], E[B1], E[B2], and E[C] ............19 3.2.Determine the optimal operating policy ...................20 4.Conclusions and Future Research ............................25 4.1.Conclusions ..............................................25 4.2.Future research ..........................................25 References ...................................................2

Management queues performances in postal traffic

U doktorskoj disertaciji rešavaju se sledeći problemi: problem opisivanja sistema masovnog opsluživanja kada teorija masovnog opsluživanja nailazi na ograničenja primene, problem predviđanja vremena čekanja, problem modelovanja odnosa na tržištu ekspres usluga kao izvora uticaja na redove čekanja, problem upravljanja brojem aktivnih kanala sistema masovnog opsluživanja i problem uticaja na subjektivno vreme čekanja. Primenom elemenata veštačke inteligencije i statističkih metoda razvijen je model za predviđanje parametra vremena čekanja u realnom vremenu pri jedinicama poštanske mreže za pružanje usluga korisnicima.The dissertation provides answers to the following issues: the problem of describing the queueing system when the queueing theory encounters limitations in its use, predicting the waiting time, the problem of modeling relations in the market of express services as a source of influence on the queues, managing the number of active channels in the queueing systems and the impact on subjective waiting time. Through application of artificial intelligence and statistical methods, a model has been developed which in real time predicts the parameters of waiting time at the units of postal network that provide service to customers