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20-389: An M/D/1 queue is a stochastic process whose state space is the set {0,1,2,3,...} where the value corresponds to the number of entities in the system, including any currently in service. The transition probability matrix for an M/D/1 queue with arrival rate λ and service time 1, such that λ <1 (for stability of the queue) is given by P as below: P = (

40-776: A 0 a 1 a 2 a 3 . . . a 0 a 1 a 2 a 3 . . . 0 a 0 a 1 a 2 . . . 0 0 a 0 a 1 . . . . . . . . . . . . . . . . . . ) {\displaystyle P={\begin{pmatrix}a_{0}&a_{1}&a_{2}&a_{3}&...\\a_{0}&a_{1}&a_{2}&a_{3}&...\\0&a_{0}&a_{1}&a_{2}&...\\0&0&a_{0}&a_{1}&...\\...&...&...&...&...\\\end{pmatrix}}} ，

60-658: A n = λ n n ! e − λ {\displaystyle a_{n}={\frac {\lambda ^{n}}{n!}}e^{-\lambda }} , n = 0,1,.... The following expressions present the classic performance metrics of a single server queuing system such as M/D/1, with: The average number of entities in the system, L is given by: L = ρ + 1 2 ( ρ 2 1 − ρ ) ; {\displaystyle L=\rho +{\frac {1}{2}}\left({\frac {\rho ^{2}}{1-\rho }}\right);} The average number of entities in

80-527: A "system". It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory . For instance, the toy problem Vacuum World has a discrete finite state space in which there are a limited set of configurations that the vacuum and dirt can be in. A "counter" system, where states are the natural numbers starting at 1 and are incremented over time has an infinite discrete state space. The angular position of an undamped pendulum

100-416: A corresponding continuous function and are therefore infinite. Discrete state spaces can also have ( countably ) infinite size, such as the state space of the time-dependent "counter" system, similar to the system in queueing theory defining the number of customers in a line, which would have state space {0, 1, 2, 3, ...}. Exploring a state space is the process of enumerating possible states in search of

120-504: A goal state. The state space of Pacman , for example, contains a goal state whenever all food pellets have been eaten, and is explored by moving Pacman around the board. A search state is a compressed representation of a world state in a state space, and is used for exploration. Search states are used because a state space often encodes more information than is necessary to explore the space. Compressing each world state to only information needed for exploration improves efficiency by reducing

140-494: Is a continuous (and therefore infinite) state space. State spaces are useful in computer science as a simple model of machines. Formally, a state space can be defined as a tuple [ N ,  A ,  S ,  G ] where: A state space has some common properties: For example, the Vacuum World has a branching factor of 4, as the vacuum cleaner can end up in 1 of 4 adjacent squares after moving (assuming it cannot stay in

160-485: Is the mean service time; σ is the variance of service time; and ρ=λτ < 1, λ being the arrival rate of the customers. For M/M/1 queue, the service times are exponentially distributed, then σ = τ and the mean waiting time in the queue denoted by W M is given by the following equation: W M = ρ τ 1 − ρ {\displaystyle {W_{M}}={\frac {\rho \tau }{1-\rho }}} Using this,

180-504: The corresponding equation for M/D/1 queue can be derived, assuming constant service times. Then the variance of service time becomes zero, i.e. σ = 0. The mean waiting time in the M/D/1 queue denoted as W D is given by the following equation: W D = ρ τ 2 ( 1 − ρ ) {\displaystyle {W_{D}}={\frac {\rho \tau }{2(1-\rho )}}} From

200-420: The next adapter to which each packet should go and dispatch the packets accordingly. Here the service time is the processing of the packet header and cyclic redundancy check, which are independent of the length of each arriving packets. Hence, it can be modeled as a M/D/1 queue. State space In computer science , a state space is a discrete space representing the set of all possible configurations of

220-414: The number of customers in the queue and mean queue length can be computed using probability generating functions . The transient solution of an M/D/1 queue of finite capacity N, often written M/D/1/N, was published by Garcia et al. in 2002. Includes applications in wide area network design , where a single central processor to read the headers of the packets arriving in exponential fashion, then computes

SECTION 10

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240-605: The number of states in the search. For example, a state in the Pacman space includes information about the direction Pacman is facing (up, down, left, or right). Since it does not cost anything to change directions in Pacman, search states for Pacman would not include this information and reduce the size of the search space by a factor of 4, one for each direction Pacman could be facing. Standard search algorithms are effective in exploring discrete state spaces. The following algorithms exhibit both completeness and optimality in searching

260-490: The queens, 92. In many games the effective state space is small compared to all reachable/legal states. This property is also observed in Chess , where the effective state space is the set of positions that can be reached by game-legal moves. This is far smaller than the set of positions that can be achieved by placing combinations of the available chess pieces directly on the board. All continuous state spaces can be described by

280-411: The queue (line), ω Q is given by: ω Q = ρ 2 μ ( 1 − ρ ) {\displaystyle \omega _{Q}={\frac {\rho }{2\mu (1-\rho )}}} Considering a system that has only one server, with an arrival rate of 20 entities per hour and the service rate is at a constant of 30 per hour. So the utilization of

300-598: The queue (line), L Q is given by: L Q = 1 2 ( ρ 2 1 − ρ ) ; {\displaystyle L_{Q}={\frac {1}{2}}\left({\frac {\rho ^{2}}{1-\rho }}\right);} The average waiting time in the system, ω is given by: ω = 1 μ + ρ 2 μ ( 1 − ρ ) ; {\displaystyle \omega ={\frac {1}{\mu }}+{\frac {\rho }{2\mu (1-\rho )}};} The average waiting time in

320-404: The queue: The busy period is the time period measured from the instant a first customer arrives at an empty queue to the time when the queue is again empty. This time period is equal to D times the number of customers served. If ρ  < 1, then the number of customers served during a busy period of the queue has a Borel distribution with parameter ρ . A stationary distribution for

340-423: The same square nor move diagonally). The arcs of Vacuum World are bidirectional, since any square can be reached from any adjacent square, and the state space is not a tree since it is possible to enter a loop by moving between any 4 adjacent squares. State spaces can be either infinite or finite, and discrete or continuous. The size of the state space for a given system is the number of possible configurations of

360-746: The server is: ρ=20/30=2/3. Using the metrics shown above, the results are as following: 1) Average number in line L Q = 0.6667; 2) Average number in system L =1.333; 3) Average time in line ω Q = 0.033 hour; 4) Average time in system ω = 0.067 hour. For an equilibrium M/G/1 queue, the expected value of the time W spent by a customer in the queue are given by Pollaczek-Khintchine formula as below: E ( W ) = ρ τ 2 ( 1 − ρ ) ( 1 + σ 2 τ 2 ) {\displaystyle E(W)={\frac {\rho \tau }{2(1-\rho )}}\left(1+{\frac {\sigma ^{2}}{\tau ^{2}}}\right)} where τ

380-481: The space. If the size of the state space is finite, calculating the size of the state space is a combinatorial problem. For example, in the Eight queens puzzle , the state space can be calculated by counting all possible ways to place 8 pieces on an 8x8 chessboard. This is the same as choosing 8 positions without replacement from a set of 64, or This is significantly greater than the number of legal configurations of

400-509: The two equations above, we can infer that Mean queue length in M/M/1 queue is twice that in M/D/1 queue. The number of jobs in the queue can be written as M/G/1 type Markov chain and the stationary distribution found for state i (written π i ) in the case D  = 1 to be Define ρ  =  λ / μ as the utilization; then the mean delay in the system in M/D/1 queue is and in

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