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Introduction to Queueing Theory and Stochastic Teletraffic Models
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1 Introduction to Queueing Theory and Stochastic Teletraffic Models Moshe Zukerman EE Department, City University of Hong Kong Copyright M. Zukerman c Preface The aim of this textbook is to provide students with basic knowledge of stochastic models that may apply to telecommunications research areas, such as traffic modelling, resource provisioning and traffic management. These study areas are often collectively called teletraffic. This book assumes prior knowledge of a programming language, mathematics normally taught in an electrical engineering course. The book aims to enhance intuitive and physical understanding of the theoretical concepts it introduces. The famous mathematician Pierre-Simon Laplace is quoted to say that Probability is common sense reduced to calculation [14]; as the content of this book falls under the field of applied probability, Laplace s quote very much applies. Accordingly, the book aims to link intuition and common sense to the mathematical models and techniques it uses. A unique feature of this book is the considerable attention given to guided projects involving computer simulations and analyzes. By successfully completing the programming assignments, students learn to simulate and analyze stochastic models, such as queueing systems and networks, and by interpreting the results, they gain insight into the queueing performance effects and principles of telecommunications systems modelling. Although the book, at times, provides intuitive explanations, it still presents the important concepts and ideas required for the understanding of teletraffic, queueing theory fundamentals and related queueing behavior of telecommunications networks and systems. These concepts and ideas form a strong base for the more mathematically inclined students who can follow up with the extensive literature on probability models and queueing theory. A small sample of it is listed at the end of this book. The first two chapters provide a revision of probability and stochastic processes topics relevant to the queueing and teletraffic models of this book. These two chapters provide a summary of the key topics with relevant homework assignments that are especially tailored for understanding the queueing and teletraffic models discussed in later chapters. The content of these chapters is mainly based on [14, 25, 73, 78, 79, 80]. Students are encouraged to study also
2 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 2 the original textbooks that include far more explanations, illustrations, discussions, examples and homework assignments. Chapter 3 discusses general queueing notation and concepts and it should be studied well. Chapter 4 aims to assist the student to perform simulations of queueing systems. Simulations are useful and important in the many cases where exact analytical results are not available. An important learning objective of this book is to train students to perform queueing simulations. Chapter 5 provides analyses of deterministic queues. Many queueing theory books tend to exclude deterministic queues; however, the study of such queues is useful for beginners in that it helps them better understand non-deterministic queueing models. Chapters 6 14 provide analyses of a wide range of queueing and teletraffic models most of which fall under the category of continuous-time Markov-chain processes. Chapter 15 provides an example of a discrete-time queue that is modelled as a discrete-time Markov-chain. In Chapters 16 and 17, various aspects of a single server queue with Poisson arrivals and general service times are studied, mainly focussing on mean value results as in [13]. Then, in Chapter 18, some selected results of a single server queue with a general arrival process and general service times are provided. Next, in Chapter 19, we extend our discussion to queueing networks. Finally, in Chapter 20, stochastic processes that have been used as traffic models are discussed with special focus on their characteristics that affect queueing performance. Throughout the book there is an emphasis on linking the theory with telecommunications applications as demonstrated by the following examples. Section 1.19 describes how properties of Gaussian distribution can be applied to link dimensioning. Section 6.6 shows, in the context of an M/M/1 queueing model, how optimally to set a link service rate such that delay requirements are met and how the level of multiplexing affects the spare capacity required to meet such delay requirement. An application of M/M/ queueing model to a multiple access performance problem [13] is discussed in Section 7.6. In Sections 8.8 and 9.5, discussions on dimensioning and related utilization issues of a multi-channel system are presented. Especially important is the emphasis on the insensitivity property of models such as M/M/, M/M/k/k, processor sharing and multi-service that lead to practical and robust approximations as described in Sections 7, 8, 13, and 14. Section 19.4 guides the reader to simulate a mobile cellular network. Section 20.6 describes a traffic model applicable to the Internet. Last but not least, the author wishes to thank all the students and colleagues that provided comments and questions that helped developing and editing the manuscript over the years.
3 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 3 Contents 1 Revision of Relevant Probability Topics Events, Sample Space, and Random Variables Probability, Conditional Probability and Independence Probability and Distribution Functions Joint Distribution Functions Conditional Probability for Random Variables Independence between Random Variables Convolution Selected Discrete Random Variables Non-parametric Bernoulli Geometric Binomial Poisson Pascal Discrete Uniform Continuous Random Variables and their Distributions Selected Continuous Random Variables Uniform Exponential Relationship between Exponential and Geometric Random Variables Hyper-Exponential Erlang Hypo-Exponential Gaussian Pareto Moments and Variance Mean (or Expectation) Moments Variance Conditional Mean and the Law of Iterated Expectation Conditional Variance and the Law of Total Variance Mean and Variance of Specific Random Variables Sample Mean and Sample Variance Covariance and Correlation Transforms Z-transform Laplace Transform Multivariate Random Variables and Transform Probability Inequalities and Their Dimensioning Applications Limit Theorems Link Dimensioning Case 1: Homogeneous Individual Sources Case 2: Non-homogeneous Individual Sources Case 3: Capacity Dimensioning for a Community
4 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 4 2 Relevant Background in Stochastic Processes General Concepts Two Orderly and Memoryless Point Processes Bernoulli Process Poisson Process Markov Modulated Poisson Process Discrete-time Markov-chains Definitions and Preliminaries Transition Probability Matrix Chapman-Kolmogorov Equation Marginal Probabilities Properties and Classification of States Steady-State Probabilities Birth and Death Process Reversibility Multi-Dimensional Markov-chains Continuous Time Markov-chains Definitions and Preliminaries Birth and Death Process First Passage Time Transition Probability Function Steady-State Probabilities Multi-Dimensional Continuous Time Markov-chains The Curse of Dimensionality Simulations Reversibility Renewal Process General Queueing and Teletraffic Concepts Notation Utilization Little s Formula Traffic Work Conservation PASTA Bit-rate Versus Service Rate Queueing Models Simulations Confidence Intervals Simulation of a G/G/1 Queue Deterministic Queues D/D/ D/D/k D/D/k/k The D/D/k/k process and its cycles
5 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman Blocking probability, mean queue-size and utilization Proportion of time spent in each state Summary of Results M/M/ Steady-State Queue Size Probabilities State Transition Diagram of M/M/ Delay Statistics Mean Delay of Delayed Customers Using Z-Transform Multiplexing Dimensioning Based on Delay Distribution The Departure Process Mean Busy Period and First Passage Time A Markov-chain Simulation of M/M/ M/M/ Offered and Carried Traffic Steady-State Equations Solving the Steady-State Equations State Transition Diagram of M/M/ Insensitivity Applications A multi-access model Birth rate evaluation M/M/k/k and Extensions M/M/k/k: Offered, Carried and Overflow Traffic The Steady-State Equations and Their Solution Recursion and Jagerman Formula The Special Case: M/M/1/ Lower bound of E k (A) Monotonicity of E k (A) The Limit k with A/k Fixed M/M/k/k: Dimensioning and Utilization M/M/k/k under Critical Loading Insensitivity and Many Classes of Customers A Markov-chain Simulation of M/M/k/k M/M/k/k with Preemptive Priorities Overflow Traffic of M/M/k/k Multi-server Loss Systems with Non-Poisson Input M/M/k Steady-State Equations and Their Solution Erlang C Formula Mean Queue Size, Delay, Waiting Time and Delay Factor Mean Delay of Delayed Customers Dimensioning
6 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman Utilization Engset Loss Formula Steady-State Equations and Their Solution Blocking Probability Obtaining the Blocking Probability by a Recursion Insensitivity Load Classifications and Definitions The Small Idle Period Limit The Many Sources Limit Obtaining the Blocking Probability by Successive Iterations State Dependent SSQ Queueing Models with Finite Buffers D/D/1/N SLB/D/1/N M/M/1/N M/M/k/N MMPP(2)/M/1/N M/E m /1/N Saturated Queues Processor Sharing The M/M/1-PS queue Insensitivity Multi-service Loss Model Model Description Attributes of the Multi-service Loss Model A Simple Example with I = 2 and k = Other Reversibility Criteria for Markov Chains Computation A General Treatment Infinite number of servers Finite Number of Servers Critical Loading Discrete-Time Queue M/G/ Pollaczek Khintchine Formula: Residual Service Approach [13] Pollaczek Khintchine Formula: by Kendall s Recursion [51] Special Cases: M/M/1 and M/D/ Busy Period M/G/1 with non-fifo disciplines M/G/1-LIFO
7 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman M/G/1 with m priority classes Nonpreemptive Preemptive Resume Queues with General Input Reich s Formula Queue Size Versus Virtual Waiting Time G/GI/1 Queue and Its G/GI/1/k Equivalent Queueing Networks Jackson Networks Computation of Blocking Probability in Circuit Switched Networks by Erlang Fixed-Point Approximation Parameter Conversion in Optical Circuit Switched Networks A Markov-chain Simulation of a Mobile Cellular Network Stochastic Processes as Traffic Models Parameter Fitting Poisson Process Markov Modulated Poisson Process (MMPP) Autoregressive Gaussian Process Exponential Autoregressive (1) Process Poisson Pareto Burst Process
8 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 8 1 Revision of Relevant Probability Topics Probability theory provides the foundation for queueing theory and stochastic teletraffic models, therefore it is important that the student masters the probability concepts required for the material that follows. We aim to provide in this chapter sufficient coverage for readers that have some probability background. Although the cover here is comprehensive in the sense that it discusses all the probability concepts and techniques used in later chapters, it does not include the many examples and exercises that are normally included in a probability textbook to help readers grasp the material better. Therefore, readers without prior probability background may be aided by additional probability texts, such as [14] and [79]. 1.1 Events, Sample Space, and Random Variables Consider an experiment with an uncertain outcome. The term experiment refers to any uncertain scenario, such as tomorrow s weather, tomorrow s share price of a certain company, or the result of flipping a coin. The sample space is a set of all possible outcomes of an experiment. An event is a subset of the sample space. Consider, for example, an experiment which consists of tossing a die. The sample space is {1, 2, 3, 4, 5, 6}, and an event could be the set {2, 3}, or {6}, or the empty set {} or even the entire sample space {1, 2, 3, 4, 5, 6}. Events are called mutually exclusive if their intersection is the empty set. A set of events is said to be exhaustive if its union is equal to the sample space. A random variable is a real valued function defined on the sample space. This definition appears somewhat contradictory to the wording random variable as a random variable is not at all random, because it is actually a deterministic function which assigns a real valued number to each possible outcome of an experiment. It is the outcome of the experiment that is random and therefore the name: random variable. If we consider the flipping a coin experiment, the possible outcomes are Head (H) and Tail (T), hence the sample space is {H, T}, and a random variable X could assign X = 1 for the outcome H, and X = 0 for the outcome T. If X is a random variable then Y = g(x) for some function g( ) is also a random variable. In particular, some functions of interest are Y = cx for some constant c and Y = X n for some integer n. If X 1, X 2, X 3,..., X n is a sequence of random variables, then Y = n i=1 X i is also a random variable. Homework 1.1 Consider the experiment to be tossing a coin. What is the Sample Space? What are the events associated with this Sample Space? Guide Notice that although the sample space includes only the outcome of the experiments which are Head (H) and Tail (T), the events associated with this samples space includes also the empty
9 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 9 set which in this case is the event {H T } and the entire sample space which in this case is the event {H T }. 1.2 Probability, Conditional Probability and Independence Consider a sample space S. Let A be a subset of S, the probability of A is the function on S and all its subsets, denoted P(A), that satisfies the following three axioms: 1. 0 P (A) 1 2. P (S) = 1 3. The probability of the union of mutually exclusive events is equal to the sum of the probabilities of these events. Normally, higher probability signifies higher likelihood of occurrence. In particular, if we conduct a very large number of experiments, and we generate the histogram by measuring how many times each of the possible occurrences actually occurred. Then we normalize the histogram by dividing all its values by the total number of experiments to obtain the relative frequencies. These measurable relative frequencies can provide intuitive interpretation to the theoretical concept of probability. Accordingly, the term limiting relative frequency is often used as interpretation of probability. We use the notation P (A B) for the conditional probability of A given B, which is the probability of the event A given that we know that event B has occurred. If we know that B has occurred, it is our new sample space, and for A to occur, the relevant experiments outcomes must be in A B, hence the new probability of A, namely the probability P (A B), is the ratio between the probability of A B and the probability of B. Accordingly, P (A B) = P (A B). (1) P (B) Since the event {A B} is equal to the event {B A}, we have that so by (1) we obtain P (A B) = P (B A) = P (B A)P (A), P (A B) = P (B A)P (A). (2) P (B) Eq. (2) is useful to obtain conditional probability of one event (A) given another (B) when P (B A) is known or easier to obtain then P (A B). Remark: The intersection of A and B is also denoted by A, B or AB in addition to A B. If events A and B are independent, which means that if one of them occurs, the probability of the other to occur is not affected, then P (A B) = P (A) (3)
10 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 10 and hence, by Eq. (1), if A and B are independent then, P (A B) = P (A)P (B). (4) Let B 1, B 2, B 3,..., B n be a sequence of mutually exclusive and exhaustive events in S, and let A be another event in S. Then, A = n (A B i ) (5) i=1 and since the B i s are mutually exclusive, the events A B i s are also mutually exclusive. Hence, n P (A) = P (A B i ). (6) Thus, by Eq. (1), P (A) = i=1 n P (A B i ) P (B i ). (7) i=1 The latter is a very useful formula for deriving probability of a given event by conditioning and unconditioning on a set of mutually exclusive and exhaustive events. It is called the Law of Total Probability. Therefore, by Eqs. (7) and (1) (again), we obtain the following formula for conditional probability between two events: P (B 1 A) = P (A B 1 )P (B 1 ) n i=1 P (A B i) P (B i ). (8) The latter is known as Bayes theorem (or Bayes law or Bayes rule). Note that Eq. (1) is also referred to as Bayes theorem. 1.3 Probability and Distribution Functions Random variables are related to events. When we say that random variable X takes value x, this means that x represents a certain outcome of an experiment which is an event, so {X = x} is an event. Therefore, we may assign probabilities to all possible values of the random variable. This function denoted P X (x) = P (X = x) will henceforth be called probability function. Other names used in the literature for a probability function include probability distribution function, probability distribution, or simply distribution. Because probability theory is used in many applications, in many cases, there are many alternative terms to describe the same thing. It is important that the student is familiar with these alternative terms, so we will use these terms alternately in this book. The cumulative distribution function) of random variable X is defined for all x R (R being the set of all real numbers), is defined as F X (x) = P (X x). (9) Accordingly, the complementary distribution function F X (x) is defined by F X (x) = P (X > x). (10)
11 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 11 Consequently, for any random variable, for every x R, F (x) + F (x) = 1. As the complementary and the cumulative distribution functions as well as the probability function can be obtained from each other, we will use the terms distribution function when we refer to any of these functions without being specific. We have already mentioned that if X is a random variable, then Y = g(x) is also a random variable. In this case, if P X (x) is the probability function of X then the probability function of Y is P Y (y) = P X (x). (11) x:g(x)=y 1.4 Joint Distribution Functions In some cases, we are interested in the probability that two or more random variables are within a certain range. For this purpose, we define, the joint distribution function for n random variables X 1, X 2,..., X n, as follows: F X1, X 2,...,X n (x 1, x 2,..., x n ) = P (X 1 x 1, X 2 x 2,..., X n x n ). (12) Having the joint distribution function, we can obtain the distribution function of a single random variable, say, X 1, as F X1 (x 1 ) = F X1, X 2,..., X n (x 1,,..., ). (13) A random variable is called discrete if it takes at most a countable number of possible values. On the other hand, a continuous random variable takes an uncountable number of possible values. When the random variables X 1, X 2,..., X n are discrete, we can use their joint probability function which is defined by P X1, X 2,..., X n (x 1, x 2,..., x n ) = P (X 1 = x 1, X 2 = x 2,..., X n = x n ). (14) The probability function of a single random variable can then be obtained by P X1 (x 1 ) = P X1, X 2,..., X n (x 1, x 2,..., x n ). (15) x 2 x n In this section and in sections 1.5, 1.6, 1.7, when we mention random variables or their distribution functions, we consider them all to be discrete. Then, in Section 1.9, we will introduce the analogous definitions and notation relevant to their continuous counterparts. We have already mention the terms probability function, distribution, probability distribution function and probability distribution. These terms apply to discrete as well as to continuous random variables. There are however additional terms that are used to describe probability function only for discrete random variable they are: probability mass function, and probability mass, and there are equivalent terms used only for continuous random variables they are probability density function, density function and simply density.
12 P Y (y) = x Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman Conditional Probability for Random Variables The conditional probability concept, which we defined for events, can also apply to random variables. Because {X = x}, namely, the random variable X takes value x, is an event, by the definition of conditional probability (1) we have P (X = x Y = y) = P (X = x, Y = y). (16) P (Y = y) Let P X Y (x y) = P (X = x Y = y) be the conditional probability of a discrete random variable X given Y, we obtain by (16) P X Y (x y) = P X,Y (x, y). (17) P Y (y) Noticing that P X,Y (x, y), (18) we obtain by (17) P X Y (x y) = 1. (19) x This means that if we condition on the event {Y = y} for a specific y, the probability function of X given {Y = y} is a legitimate probability function. This is consistent with our discussion above. The event {Y = y} is the new sample space and X has a legitimate distribution function there. By (17) P X,Y (x, y) = P X Y (x y)p Y (y) (20) and by symmetry P X,Y (x, y) = P Y X (y x)p X (x) (21) so the latter and (18) gives P Y (y) = x P X,Y (x, y) = x P Y X (y x)p X (x) (22) which is another version of the Law of Total Probability (7). 1.6 Independence between Random Variables The definition of independence between random variables is very much related to the definition of independence between events because when we say that random variables U and V are independent, it is equivalent to say that the events {U = u} and {V = v} are independent for every u and v. Accordingly, random variables U and V are said to be independent if P U,V (u, v) = P U (u)p V (v) for all u, v. (23) Notice that by (20) and (23), we obtain an equivalent definition of independent random variables U and V which is P U V (u v) = P U (u) (24) which is equivalent to P (A B) = P (A) which we used to define independent events A and B.
13 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman Convolution Consider independent random variables V 1 and V 2 that have probability functions P V1 (v 1 ) and P V2 (v 2 ), respectively, and their sum which is another random variable V = V 1 + V 2. Let us now derive the probability function P V (v) of V. P V (v) = P (V 1 + V 2 = v) = v 1 P (V 1 = v 1, V 2 = v v 1 ) = v 1 P V1 (v 1 )P V2 (v v 1 ). The latter is called the convolution of the probability functions P V1 (v 1 ) and P V2 (v 2 ). Let us now extend the result from two to k random variables. Consider k independent random variables X i, i = 1, 2, 3,..., k. Let P Xi (x i ) be the probability function of X i, for i = 1, 2, 3,..., k, and let Y = k i=1 X i. If k = 3, we first compute the convolution of X 1 and X 2 to obtain the probability function of V = X 1 + X 2 using the above convolution formula and then we use the formula again to obtain the probability function of Y = V +X 3 = X 1 +X 2 +X 3. Therefore, for an arbitrary k, we obtain ( ) k P Y (y) = P X1 (y Σ k i=2x i ) P Xi (x i ). (25) x 2, x 3,..., x k : x 2 +x x k y If all the random variable X i, i = 1, 2, 3,..., k, are independent and identically distributed (IID) random variables, with probability function P X1 (x), then the probability function P Y (y) is called the k-fold convolution of P X1 (x). i=2 Homework 1.2 Consider again the experiment to be tossing a coin. Assume that P (H) = P (T ) = 0.5. Illustrate each of the Probability Axioms for this case. Homework 1.3 Now consider an experiment involving three coin tosses. The outcome of the experiment is now a 3-long string of Heads and Tails. Assume that all coin tosses have probability 0.5, and that the coin tosses are independent events. 1. Write the sample space where each outcome of the experiment is an ordered 3-long string of Heads and Tails. 2. What is the probability of each outcome? 3. Consider the event Find P (A) using the additivity axiom. A = {Exactly one head occurs}.
14 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 14 Partial Answer: P (A) = 1/8 + 1/8 + 1/8 = 3/8. Homework 1.4 Now consider again three coin tosses. Find the probability P (A B) where A and B are the events: A = more that one head came up B = 1st toss is a head. Guide: P (B) = 4/8; P (A B) = 3/8; P (A B) = (3/8)/(4/8) = 3/4. Homework 1.5 Consider a medical test for a certain disease. The medical test detects the disease with probability 0.99 and fails to detect the disease with probability If the disease is not present, the test indicates that it is present with probability 0.02 and that it is not present with probability Consider two cases: Case a: The test is done on a randomly chosen person from the population where the occurrence of the disease is 1/ Case b: The test is done on patients that are referred by a doctor that have a prior probability (before they do the test) of 0.3 to have the disease. Find the probability of a person to have the disease if the test shows positive outcome in each of these cases. Guide: A = person has the disease. B = test is positive. Ā = person does not have the disease. B = test is negative. We need to find P (A B). Case a: We know: P (A) = P (Ā) = P (B A) = P (B Ā) = By the Law of Total Probability: P (B) = P (B A)P (A) + P (B Ā)P (Ā). P (B) = =
15 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 15 Now put it all together and use Eq. (2) to obtain: P (A B) = Case b: P (A) = 0.3. Repeat the previous derivations to show that for this case P (A B) = Homework 1.6 In a multiple choice exam, there are 4 answers to a question. A student knows the right answer with probability 0.8 (Case 1), with probability 0.2 (Case 2), and with probability 0.5 (Case 3). If the student does not know the answer s/he always guesses with probability of success being Given that the student marked the right answer, what is the probability he/she knows the answer. Guide: A = Student knows the answer. B = Student marks correctly. Ā = Student does not know the answer. B = Student marks incorrectly. We need to find P (A B). Case 1: We know: P (A) = 0.8. P (Ā) = 0.2. P (B A) = 1. P (B Ā) = By the Law of Total Probability: P (B) = = P (B) = P (B A)P (A) + P (B Ā)P (Ā). Now put it all together and use Eq. (2) to obtain: Case 2: P (A B) = Repeat the previous derivations to obtain: P (A) = 0.2
16 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 16 P (B) = 0.4 P (A B) = 0.5. Case 3: Repeat the previous derivations to obtain: P (A) = 0.5 P (B) = P (A B) = 0.8. Homework 1.7 Watch the following youtube link on the problem known as the Monty Hall problem: https://www.youtube.com/watch?v=mhlc7peglgg Then study the Monty Hall problem in: and understand the different solution approaches and write a computer simulation to estimate the probability of winning the car by always switching. 1.8 Selected Discrete Random Variables We present here several discrete random variables and their corresponding distribution functions. Although our cover here is non-exhaustive, we do consider all the discrete random variables mentioned later in this book Non-parametric Discrete non-parametric random variable X is characterized by a set of n values that X can take with nonzero probability: a 1, a 2,..., a n, and a set of n probability values p 1, p 2,... p n, where p i = P (X = a i ), i = 1, 2,... n. The distribution of X in this case is called a non-parametric distribution because it does not depend on a mathematical function that its shape and range are determined by certain parameters of the distribution Bernoulli We begin with the Bernoulli random variable. It represents an outcome of an experiment which has only two possible outcomes. Let us call them success and failure. These two outcomes are mutually exclusive and exhaustive events. The Bernoulli random variable assigns the value X = 1 to the success outcome and the value X = 0 to the failure outcome. Let p be the probability of the success outcome, and because success and failure are mutually exclusive and exhaustive, the probability of the failure outcome is 1 p. The probability
17 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 17 function in terms of the Bernoulli random variable is: P (X = 1) = p (26) P (X = 0) = 1 p Geometric The geometric random variable X represents the number of independent Bernoulli trials, each of which with p being the probability of success, required until the first success. For X to be equal to i we must have i 1 consecutive failures and then one success in i independent Bernoulli trials. Therefore, we obtain P (X = i) = (1 p) i 1 p for i = 1, 2, 3,.... (27) The complementary distribution function of the geometric random variable is P (X > i) = (1 p) i for i = 0, 1, 2, 3,.... The geometric random variable possesses an important property called memorylessness. particular, discrete random variable X is memoryless if In P (X > m + n X > m) = P (X > n), m = 0, 1, 2,..., and n = 0, 1, 2,... (28) The Geometric random variable is memoryless because it is based on independent Bernoulli trials, and therefore the fact that so far we had m failures does not affect the probability that the next n trials will be failures. The Geometric random variable is the only discrete random variable that is memoryless Binomial Assume that n independent Bernoulli trials are performed. Let X be a random variable representing the number of successes in these n trials. Such random variable is called a binomial random variable with parameters n and p. Its probability function is: ( ) n P (X = i) = p i (1 p) n i i = 0, 1, 2,..., n. i Notice that a Binomial random variable with parameters 1 and p is a Bernoulli random variable. The Bernoulli and binomial random variables have many applications. In particular, it is used as a model for voice as well as data sources. Such sources alternates between two states on and off. During the on state the source is active and transmits at the rate equal to the transmission rate of its equipment (e.g. a modem), and during the off state, the source is idle. If p is the proportion of time that the source is active, and if we consider a superposition of n independent identical sources, than the binomial distribution gives us the probability of the number of sources which are simultaneously active which is important for resource provisioning.
18 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 18 Homework 1.8 Consider a state with voter population N. There are two candidates in the state election for governor and the winner is chosen based on a simple majority. Let N 1 and N 2 be the total number of votes obtained by Candidates 1 and 2, respectively, from voters other than Johnny. Johnny just voted for Candidate 1, and he would like to know the probability that his vote affects the election results. Johnny realizes that the only way he can affect the result of the election is if the votes are equal (without his vote) or the one he voted for (Candidate 1) had (without his vote) one call less than Candidate 2. That is, he tries to find the probability of the event 0 N 1 N 2 1. Assume that any other voter (excluding Johnny) votes independently for Candidates 1 and 2 with probabilities p 1 and p 2, respectively, and also that p 1 + p 2 < 1 to allow for the case that a voter chooses not to vote for either candidate. Derive a formula for the probability that Johnny s vote affects the election results and provide an algorithm and a computer program to compute it for the case N = 2, 000, 000 and p 1 = p 2 = 0.4. Guide By the definition of conditional probability, P (N 1 = n 1, N 2 = n 2 ) = P (N 1 = n 1 )P (N 2 = n 2 N 1 = n 1 ) so ( ) ( ) N 1 P (N 1 = n 1, N 2 = n 2 ) = p n 1 1 (1 p 1 ) N n 1 1 N n1 1 p n 2 2 (1 p 2 ) N n 1 1 n 2. n 1 n 2 Then, as the probability of the union of mutually exclusive events is the sum of their probabilities, the required probability is (N 1)/2 k=0 P (N 1 = k, N 2 = k) + (N 1)/2 1 k=0 P (N 1 = k, N 2 = k + 1). where x is the largest integer smaller or equal to x. and x is the smallest integer greater or equal to x. Next, let us derive the probability distribution of the random variable Y = X 1 + X 2 where X 1 and X 2 are two independent Binomial random variables with parameters (N 1, p) and (N 2, p), respectively. This will require the derivation of the convolution of X 1 and X 2 as follows. P Y (k) = P (X 1 + X 2 = k) k = P ({X 1 = i} {X 2 = k i}) = i=0 k P X1 (i)p X2 (k i) i=0
19 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 19 k ( ) ( ) N1 = p i (1 p) N N2 1 i p k i (1 p) N 2 (k i) i k i i=0 k ( )( ) = p k (1 p) N 1+N 2 k N1 N2 i k i i=0 ( ) = p k (1 p) N 1+N 2 k N1 + N 2. k We can conclude that the convolution of two independent binomial random variables with parameters (N 1, p) and (N 2, p) has a binomial distribution with parameters N 1 + N 2, p. This is not surprising. As we recall the binomial random variable represents the number of successes of a given number of independent Bernoulli trials. Thus, the event of having k Bernoulli successes out of N 1 + N 2 trials is equivalent to the event of having some (or none) of the successes out of the N 1 trials and the remaining out of the N 2 trials. Homework 1.9 In the last step of the above proof, we have used the equality Prove this equality. ( ) N1 + N 2 = k k i=0 ( N1 )( ) N2. i k i Guide Consider the equality (1 + α) N 1+N 2 = (1 + α) N 1 (1 + α) N 2. Then equate the binomial coefficients of α k on both sides Poisson A Poisson random variable with parameter λ has the following probability function: λ λi P (X = i) = e i! i = 0, 1, 2, 3,.... (29) To compute the values of P (X = i), it may be convenient to use the recursion with P (X = i + 1) = P (X = 0) = e λ. λ P (X = i) (30) i + 1
20 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 20 However, if the parameter λ is large, there may be a need to set ˆP (X = λ ) = 1, where x is the largest integer smaller or equal to x, and to compute recursively, using (30) (applied to the ˆP (X = i) s), a sufficient number of ˆP (X = i) values for i > λ and i < λ such that ˆP (X = i) > ɛ, where ɛ is chosen to meet a given accuracy requirement. Clearly the ˆP (X = i) s do not sum up to one, and therefore they do not represent a probability distribution. To approximate the Poisson probabilities they will need to be normalized as follows. Let α and β be the lower and upper bounds, respectively, of the i values for which ˆP (X = i) > ɛ. Then, the probabilities P (X = i) are approximated using the normalization: P (X = i) = { ˆP (X=i) β i=α ˆP (X=i) α i β 0 otherwise. The importance of the Poisson random variable lies in its property to approximate the binomial random variable in case when n is very large and p is very small so that np is not too large and not too small. In particular, consider a sequence of binomial random variables X n, n = 1, 2,... with parameters (n, p) where λ = np, or p = λ/n. Then the probability function (31) lim P (X n = k) n is a Poisson probability function with parameter λ. To prove this we write: Substituting p = λ/n, we obtain ( ) n lim P (X n = k) = lim p k (1 p) n k. n n k or lim P (X n! n = k) = lim n n (n k)!k! ( ) k ( λ 1 λ n k. n n) Now notice that and lim P (X n! n = k) = lim n n (n k)!n k ( λ k k! ( lim 1 λ n = e n n) λ, ( lim 1 λ k = 1, n n) ) ( 1 λ ) n ( 1 λ k. n n)
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Let us know when your receive great advice so we can thank our lawyers and point others with similar questions in the right direction.Got it!
An officer has been telling people that I have a drug house & they shouldn't be coming here. He actually told this to my friend, in front of his daughter who in turn told my daughter what was said. Now my daughter is asking why a cop would say this if it wasn't true & I have no explanation because it's not true! I've always taught my kids to trust cops, so now what? I haven't touched drugs since 2002. This isn't the first time this has happened with this officer either. It's to the point that some of the neighborhood kids can't play anymore because of the rumors this cop is telling. I was even investigated by DCFS recently but passed all of their drug tests on the spot. The case was dropped, but the cop is still pulling people over when they leave here & telling them lies.
maybe, but I am not seeing a particularly good case for slander. It sounds like there is some basis for his allegations, i.e. "2002", however, in order for him to pull people over he must have a reasonable suspicion that the person has been, is, or is about to be engaged in criminal activity. If this is an ongoing harassment you might want to consider a departmental complaint, as a first step to a potential lawsuit against both the officer and the department.
The biggest problems I see in your case are both liability, and damages. That means it is likely not a great case, as those are two of the cornerstones in any case of this type. I would recommend that you keep a record of when the officer makes these stops, what is said, and how you know, in an attempt to help overcome your liability issues. As to damages you are going to have to quantify how you have been harmed in a monetary fashion. In your description it sounds like there are no quantifiable damages yet. A better case from a damages perspective is one where you feel compelled to send the children, and perhaps yourself, to counseling in attempts to deal with the officer's actions.
Good luck, I hope you are able to get the officer to leave you alone.Ask a similar question
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subset: commoncrawl


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Hot Chip - My Piano (Dub) MP3
This is the flip side to the single "My Piano." It was released for the ever amazing DJ-Kicks series. It's a different take from the original, but still very good... enjoy!
Hot Chip - My Piano (dub) MP3
From the "DJ-Kicks - My Piano" EP Info: http://www.discogs.com/Hot-Chip-My-Piano/release/2333624 Label: Studio !K7 Recorded in 2007 from the "diabolika" ...
Piano B - Dub My Roots MP3
Roots of Dub Funk 4: Rise of the Electric Dread, track 13, as always please support the Artist and buy a copy of the album.
Dell Gibbons vs Miss Roberts - Freedom (Fonzerelli's Superclub Piano Dub) MP3
d[-_-]b · · · · · https://www.facebook.com/88Excalibur88Music.
CHOCOLAT (Piano Dub) MP3
My version of CHOCOLAT. Music&Lyrics by TYC FORCE Piano by celestialdreams31 Yeah! Hopping on the bandwagon! But I really love this song.. this episode ...
Lost My Music Piano English Dub | The Melancholy of Haruhi Suzumiya | StellTsirlis Cover MP3
I OWN NOTHING BUT MY VOICE!!! Bobby Thong wrote these lovely lyrics! I believe :3 bubbawaba98 provided the piano! And Haruhi is property of Nagaru ...
Sunday Night Dub (Live Piano Dubstep) MP3
This is a dubstep jam. I programmed the drums in Reason. My piano is routed through a Korg AX1B for a mild distortion sound (used for the basslines), then an ...
His Eye Is On The Sparrow [piano dub-lydia paek] MP3
Lydia Paek is one of my favorite Youtube vocalist. :) Her voice is amazingly soulful and raw. She obviously doesn't need any accompaniment, but that's what i do ...
DELL GIBBONS feat. MISS ROBERTS - Freedom (Fonzerelli's Superclub Piano Dub Edit) MP3
McM ::.
MLP Equestria Girls - Rainbow Rocks - Player Piano - Dub PL MP3
Tytuł: Pianistka Wszelkie prawa należą do Hasbro Pliki od Galapagos.
Native Sons & Inaya Day - City Life (Louie Vega Piano Dub) MP3
Label: http://www.vegarecords.net | 23.12.2013 Download: http://shop.vegarecords.net/collections/view-all-releases/products/city-life-louie-vega-remix ...
Helux - Piano Dub (Dubstep) MP3
My newest tune. Please rate, comment, and subscribe.
Angelo Bavaro Producer - Between Me And You (Piano Dub Mix) MP3
Download this song: https://itunes.apple.com/us/album/between-me-and-you-single/id971244316 ------------------------------- Music and Lyrics: Angelo Bavaro.
Boso - Piano Dub ( Official Music ) MP3
My name is Boso' . I'm an amateur music producer who lgets bored a lot . I'm specialized...
Dub Inc - Rude boy piano MP3
Dub Incorporation - Rudeboy (piano cover by Chocolate Bear) (by ear, no sheets :s) my facebook: http://www.facebook.com/?ref=home#!
Soulful House: The Rurals - Lush (Kwame's Piano Dub) MP3
The Rurals - Lush (Kwame's Piano Dub) Taken from the Album "Reinterpretations" Peng Records, 2003 Buy Link: ...
Lindsey Stirling's "Transcendence" Piano dub/cover MP3
MY VERY FIRST YOUTUBE SUBMISSION!!! This is a version that I did of Lindsey Stirling's track "Transcendence." Subscribe to Lindsey's channel here!
Old Skool Piano Classics Vol 3 (Mixed By DJ C-Dub) MP3
Vol 3 is here.....As always this mix has been put together by using some of my best and most loved old skool records! All tunes featured in this mix can be found ...
Hamdiman-Piano Dub MP3
FREE DOWNLOAD LINK:http://soundcloud.com/hamdiman/hamdiman-piano-dubstep Background by:http://www.youtube.com/user/antijuicemonkey2 This is my ...
Ordinary People [piano dub-lydia paek] MP3
lydia paek singing "ordinary people" My favorite Youtube vocalist. :) awesome awesome. :) Dubbed piano over it just for fun. apologies, it's just a tad loud. Lydia ...
Dubstep Piano on the lake - Radioactive - With William Joseph - 4K MP3
Had the chance to work with the amazing William Joseph on his take of Radioactive by Imagine Dragons. You can buy this song on iTunes: ...
Baaba Maal - Gorel (MAW Piano Dub) MP3
African singer Baaba Maal remixed by Masters At Work in 1995 on the UK Mango Records Label.Couldn't find this "Piano Dub Mix" on youtube so thought i'd ...
Audio Playground - Emergency (Feat. Snoop Dogg) [Chris Sammarco Frantic Piano Dub] MP3
Check out the Chris Sammarco Frantic Piano Dub Of "Emergency" by Audio Playground ft. Snoop Dogg! Want some more Emergency Remixes?
Alex White - Piano Dub (Demo) [Official] MP3
My dubstep demo: Piano Dub!!! I hope you enjoy it! Song made by Alex White! Like, share, comment and subscribe! :) Facebook: ...
FPM (Fantastic Plastic Machine) / Forever Mine (Piano Dub) (2009 "FPM") MP3
FPM (Fantastic Plastic Machine) / Forever Mine (Piano Dub) (2009 ""FPM"") iTunes | http://apple.co/1GJxmoF mu-mo | http://bit.ly/1IAmJph 1. If you do, I do (威風 ...
Pet Shop Boys - Left To My Own Devices (Frankie Knuckles Royal Piano Dub) MP3
Unreleased version by Frankie Knuckles intended to be included in the second 12" for US which was discarded (1988). Superb orchestral intro... Enjoy! :)
My Sassy Girl Tagalog Dubbed Dvd Rip MP3
I do not own this. NO COPYRIGHT INFRINGEMENT INTENDED. Enjoy :D.
East West Connection - The More I Get...(Piano Dub) MP3
from " The More I Get..." (1999)
Spook - SR No Piano Dub MP3
Provided to YouTube by Warner Music Group Spook - SR No Piano Dub · Sean Random My House Is Your House ℗ 2011 Nervous Inc Released on: ...
Old Skool Piano Classics Vol 1 (Mixed By DJ C-Dub) MP3
Iv mixed together some of the best old skool piano classics which have been uploaded on to my YouTube channel over the years. Thanks to all those people out ...


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- HubPages»
- Technology»
- Computers & Software»
- Computer Software
What is the Best & Fastest Browser? Internet Browser Reviews: 2017
The difference between a "browser" and a "search engine".
In response to the video above. A "browser" and "a search engine" are two different things. a "browser" is one of the following... Firefox, Internet Explorer, Opera, Safari or Google Chrome etc. These are not to be confused with the hundreds of different 'search engines' you could be using such as Google, which is the most popular and best, followed by, Bing and Yahoo which are very good followed by Ask and AOL which are not my cup of tea.
Note: You can use any search engine with any browser you like.
Browser choices for PC, Smart Phones, Tablet, Notebook, Notebook and Android devices. See source
What exactly is the best and fastest browser? The many things to take into consideration...
Browsers are like vehicles: you want to travel fast, securely and in style with lots of cool features. Test drive new browsers and observe their performances. The browser I use depends on my journey. A change of browser can result in better performance, faster page loading and videos that buffer seamlessly. If you do decide to use different browsers, you'll soon notice how different they each perform, from surfing the web, plug-ins, extensions, page scrolling to variations in performances and more.
(Top 5 browser reviews and poll further down this page) Are you looking for browsers designed for smart phones and other devices? If so, click here for Full Browser Listings.
What's good for the goose may not be good for the gander. Let's say John and Jane test drive every available browser. John may decide that Firefox is the best browser while Jane is sure that Internet Explorer is the best. Who is correct? Answer: Both! That's because the best browser is the one that meets their needs and more importantly, their device.
It's all about individual preference – for example, if every member of your household had the same PC and specifications, they would still never agree on which browser is the best. Why? Because each PC will have different amounts of hard drive space/capacity/shortage, etc. Each PC will have their desktop customized with fancy visual gimmicks and so on. The add-ons, and tool-bars each has, as well as many other factors all determine the speed of any browser on your computer.
One of the most common browser used today is Internet Explorer. Many people think there are no others - but there are and much better. So, what exactly is the best and fastest browser? These questions are very common, an d as you can see, have many answers...
Having a variety of browsers installed makes using the Internet easier. Because all browsers are not compatible with every website you visit. Most times you won't even realize what your not seeing. Each browser has tools and other gimmicks that are unique which makes life surfing much quicker and slicker..
Whatever new browsers you decide to use, never un-install browsers that you don't use. It's best t to keep them for a rainy day especially when you're have problems with the new browsers you're using. Having, and using different browsers on your PC is not a problem provided you have enough disc space.
What is the best browser for a Mac: Camino, Firefox, Flock and Opera all perform well on Mac machines, but overall the best browser for Mac and IOS users is Safari. Again, it's all about user preference. Try them and see which one's you prefer. They're all free! Over 100 Browsers Listed Here. Includes safe browsers designed for children to use safely.
What's best? Don't waste your time with browser comparison websites. Only YOU can decide and compare by taking them for a spin..
Firefox Reviews
Firefox has improved it's tabbed interface, pop-up blocker/built-in multiple search tools and built-in RSS reader. The privacy mode is very private. If you need an extension for something, the Firefox extension capabilities are extensive and the many themes are easy to use. It has enhanced security features to protect you against all kinds of malware, spy ware and phishing sites and handles very nicely. It's easy for all ages to use and is a very stable browser. Firefox, as well as all other browsers can be downloaded, installed and run in conjunction with any existing browsers you have.
Chat via video with anyone even if they don't use Firefox with "Firefox hello. You will need to register. More info here. https://support.mozilla.org/en-US/products/firefox/firefox-hello-webrtc
- Supports all Windows 98 to XP & Vista: Apple Mac OS X and several varieties of Linux and Red Hat Linux 8.0. Firefox is also available for the Web / Some mobile devices / android devices.
More in formation and download Firefox: https://www.mozilla.org/en-GB/firefox/products/
Internet Explorer's Edge Browser
Edge Reviews - previously called Windows Internet Explorer
Edge from Microsoft available for Windows 10. It has had major improvements than it's predecessor and about time too!. With Edge You'll notice a better design, improved tracking protection, tabbed browsing and major improvements to its security and all overall features although it lacks the array of extensions which Firefox and Chrome have.
Supports: Windows XP Service Pack 2 (SP2) Windows XP Professional x64 Edition Windows Server 2003 Service Pack 1 (SP1) - Vista compatible - Mac.
- Link to more information & Download Internet Explorer versions 7, 8, 9, 10 and 11. http://www.microsoft.com/en-gb/download/internet-explorer.aspx
- Link to more information about Windows new Edge browser. https://www.microsoft.com/en-gb/windows/microsoft-edge
Opera Reviews
Opera is a highly respected Web browser from Norway that competes very well among its competitors. It offers many unique features such as using Opera with your voice. Its mouse gestures always make navigating back and forward pages handy. The thousands of widgets and third party extensions are very useful and cool. The progress bar on the address bar lets you see pages loaded from the site. The Unite feature that allows users to file share is very impressive. More importantly, Opera has improved its security, allowing a safer and more stable surfing experience. Overall, it's a very good looking browser, performs very well and is a must have in your arsenal of browsers.
- Supports: Windows / Solaris / Intel / QNX / Mac OS / Linux / FreeBSD / BeOS. Opera is also available for PC / Tablet / Mac / Blackberry / Windows phone / ipad / Kindle Fire & other mobile devices..
Safari Reviews
Safari Web browser developed by Apple, Inc is not the most popular browser, but it does have a nice interface and a decent number of extensions to boot. Sending email links or live Web pages couldn't be easier. The history function is very helpful as it uses (iTunes style) cover-flow to scan through large thumbnails of the pages you have visited. It's particularly useful when you can't remember the URL but you can remember an image or what it looks like. It is limited in customization features, and trying to bookmark a Web page could be easier. I believe Safari could do much better especially being an "Apple" product. Further development and more time could make Safari a better browser and could be a contender in the browser wars. Safari works on Windows, Macs and other applications such as Ipad and mobile devices.
- Supports Mac and Windows: Safari is also available for PCs / Tablet's / Android / ios 7 / iphone / ipod touch.
Google Chrome
Google Chrome Reviews
Since Google released its own Web browser "Google Chrome" has become very popular and a firm favorite browser among many PC users. Its impressive speed, cool features and security optimizes its chance to be the most popular and widely used browser today.
- Supports: Windows XP, Vista and 7
Browsers designed specifically for Windows, Android, BlackBerry, ipad and iphones.
Share - Like - Tweet - Thumbs up If you found this page useful.
Which browser do you prefer?
- 100 + Browser downloads Here. All Browser Downloads. For all Smart-phones, including ipad and iPod. All Android, Mac, Window's and other mobile devices.
- What is the Best Anti-Virus Software. Reviews of the top 5 free and non free versions. Plus, Tips to help you decide which one to choose.
© 2007 compu-smart
I love Firefox, too. Great hub! I've never used Opera, but Internet Explorer has given me many problems in my PC days...
I agree that Firefox is the best of the big three, but I prefer Opera to IE. But, then I haven't seen version 7 yet.
Great hub, compu-smart. I agree Firefox is the best, and lots of other hubbers say the same thing. Good explanation of how the three big browsers compare.
I agree with your choice of Firefox, but I would rank Opera above IE. Great hub.
Of the three biggies I'd have to say that Firefox outshines the others. I use it on Mac and PC and it performs equally good on both platforms. Microsoft have stopped developing IE for Mac, so the main choice for Mac users is Safari or Firefox, and Firefox for Mac is eating into Safari's lead!
Really liked your hub - very well written.
I'm been having a love affair with Firefox for quite awhile now. Best browser on the planet. Thanks for the informative Hub.
I think you'll find many internet-security auditing firms choose Opera over the other two. Opera is, by far, the most secure of the three browsers.
Furthermore, third-party extensions are also available in Opera, however once you start to load many, it doesn't get as bloated and as slow as Firefox.
Useful hub, thank you.
I use Opera, it takes care of many trojans and viruses unlike IE. Also have firefox, but am so used to Opera, that I don't use FF much.
Firefox is great, but I also use a browser called Crazy Browser. It is older, but one of the most useful - especially in terms of how the tabbed browsing works.
Great hub, Comp! I use IE but didn't like 7 so I went back to 6. I have been considering adding FF and Opera so I think now I will just add them and do a comparison to see which one fits my needs the best. Thanks for the boost! Should have done this a long time ago!
I love Opera too, but eventually had to switch to Firefox because there are too many sites that don't work well with it.
Firefox seems just as good. I'd rate IE7 a long way third.
I use Firefox and recently downloaded the Firefox 3 Beta 5 program. I can vouch that it does have increased performance than the previous versions. I only use I.E.7 when there are compatibility issues with Firefox (which is incredibly rare for me to find any!)
Anyways, good hub!
Thx for the great info. I know Firefox is better than IE, but I still almost always use IE out of habit. I've never heard of Opera before, goodie - something new!
Thx, you've done it again, CS!
I find FIrefox goes a bit slow these days... which is irritating when you have a quad core processor and loadsof ram : (
still browser of choice though.. couldn't do without all the different plugins it has
YOu have 5 browsers on your PC? I only have 2..FF and IE. These computer thing makes my head spin. Hehe My friend is the one who installed it all here. Hmmm...maybe I should ask you to "look" into my computer too. :)
I would have to agree, firefox hits the top. theres still a lot to be done but its not doing too bad. Internet Explorer could have so much more and could work much better but it doesn't.. Safari usually does the job but theres still things firefox will display that it won't.
I have picked the best then because whilst I have Internet Explorer it hardly ever gets used anymore and I stick with Mozilla Firefox.
I experimented with FF3 and i dont have time to detail all the positive/negativespoints with its predecessor. but lets just say, im sticking with FF2, just as much as im sticking with XP over Vista- nuff said. peace
Got to love Opera. IE is definately the worst!
I'm trying to get used to Flock but still prefer Camino really.


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Turning Cooking Nightmares
into Culinary Dreams!
There are lots of ways to enjoy what we offer!
Fresh local ingredients makes for a delicious meal! They also have classes for children!Brian T.
Great food and awesome owners! Very helpful and super friendly!Elise B.
I love that I can call Dede, tell her what I need, and she is able to customize a meal to suit whatever size group or function. I've never had any food that wasn't delicious, but the salads and homemade dressings are particularly good.Andy M.
I love this place! The food is great and the service is outstanding. You should try it.Diane H.
Freshly prepared daily by Chef Dede, our Delights-to-Go menu changes on a weekly basis. We typically have available on our weekly Delights-to-Go menu three entree items, a soup, chili, or stew, Build-Your-Own salads, and at least two Specialty Salads (Chicken salads, Peppadew Cheese, and Tuna salad, et. al), some kind of dessert, and various vegetables to make your meals "Complete," if so desired. Chef Dede makes our breads, sauces, and salad dressings homemade. She uses high quality ingredients, prepared to exacting standards. We believe you'll be able to taste the difference in the menu items we offer!
Culinary Dreams caters lunches, dinners, work parties, birthday parties, family reunions, and all types of special occasions. We will adjust our menu to meet your needs, whether it be for dietary restrictions or just personal preference. Let us work with you on your next "event" by giving Chef Dede a call…251.621.0026.
Using the same fresh, high-quality ingredients as our Delights-to-Go, Chef Dede prepares our Frozen-to-Go menu items on a "batch" basis. Just as with our Delights-to-Go menu items, our Frozen-to-Go menu items also come in single serving, two serving, and four serving sizes. We always have several varieties of soups available each week, along with a nice selection of entree items in each of our categories: meatless, with chicken, with beef, and with seafood. On occasion, we have pork dishes available.
Cooking Classes & Camps
All classes are very "hands on" not just listening to a chef lecture! You will have the chance to prepare your menu items on our commercial induction cooktops and in our convection oven. All participants should wear comfortable clothes, closed-toe shoes, and pull their hair back if longer than shoulder length. We will let you use one of our aprons, and if you desire to wear a chef's hat, we have those for you as well.
Lunch Menu
We'd love for you to dine with us, but our lunches are always prepared in such a way that you can take them with you if you need to. All our menu items are prepared "to order" so we ask that if you are in a hurry, please give us a call in advance to place your lunch order. On our lunch menu, we offer Hot Lunches, several different combos, such as Soup (or Wrap) and Build-Your-Own Salad, and our popular "Roll It or Bowl It" options. If you're craving a hot sandwich, our Fresh Turkey Burgers, and 100% Grass-fed Beef Burgers on Homemade Buns built just how you like it will "hit the spot."
Private Parties
Culinary Dreams specializes in cooking parties. Chef Dede has plenty of cooking party menu suggestions, or you can work with her to create your own menu. Whether it's a Birthday Party, Bridal Shower, Baby Shower, or just some friends who want to share a great time cooking together, Culinary Dreams is the perfect place for a private cooking party!
You can taste the difference!
Because we use fresh ingredients, and make so many of our menu items from scratch, including our sauces, vinaigrettes, breads, and desserts, as well as locally sourcing as many of our vegetables as possible and using hydroponically grown lettuce from Baldwin County, we're confident you'll put Culinary Dreams on your "map" once you've given us a try.
Culinary Dreams, Inc.
2101-D U.S. Highway 98
Daphne, Alabama 36526
Store Hours
Monday-Friday: 10:30 a.m. to 6:00 p.m.
Friday Evenings & Saturday:
Private Cooking Parties By Reservation Only
© 2017 Culinary Dreams, Inc.
We hope to see you at
Culinary Dreams soon!


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An ideal trend is a series of consecutive price bars that move in the same direction in every aspect. In an ideal uptrend, every open, close, high, and low is higher than the previous one. In an ideal downtrend, every open, close, high, and low is lower than the previous one.
Especially swing traders need to pay attention to every bar to determine whether the current movement has still enough energy left or whether it will end and reverse soon.
Continue reading below the table…
[table id = 2/]
Evaluating trends
As with any aspect of trading, reality leaves little room for perfection. When you find a trend, there will be a number of stray periods that move in the wrong direction. Sometimes, these periods will be a mere result of insignificant market movements. Sometimes, however, they will be a legitimate indication of a weakening movement.
Your job as a trader is to determine one situation from the other in real-time trading. This job is not easy, and especially new binary options traders often feel nervous when one bar does not fit their prediction, which will cause them to make bad decisions.
To successfully determine whether a trend and a movement are still intact and valid, you need to use your judgment in several ways:
1) Significance
There are significant market movements, and there are market movements that mean nothing. You should have a rule that clearly defines what significance means. As a swing trader, you could get out of any movement, as soon as one period closes a certain percentage higher (in a downtrend) or lower (in an uptrend) than the preceding period.
How far does a single bar have to move for you to stop trusting the movement? Which percentage you choose should depend on your time frame (the shorter the time frame, the smaller the percentage) and your experience with the asset. If you want to automate the process, you can use momentum indicators such as the average true range or Bollinger bands.
As a trend follower, you need a similar rule. If an uptrend creates a high that is lower than the preceding high, do you stop investing, or do you wait for the trend to break the trend line or the previous point 2? Having a clearly defined rule will help you make an easy decision when the situation arises.
2) Majority
In an upwards movement, not every bar will be a perfectly rising bar. Some bars will even be perfect down bars. Still, most bars should be rising bars. As a swing trader you have to define a certain amount of directional valid bars every movement needs to have for you to invest in it.
If the movement has too many outliers, it is probably not a good fit to be traded with your strategy and you should ignore it. Similarly, trends will not always move perfect. They will create new point 3 that have not moved further than the previous point 2, and sometimes they create a new point 2 before creating a new point 3.
As trader you have to be aware of that and define an amount of imperfection you allow every trend to have. As long as the trend moves within these limits, you can still trust it and invest in it. If it has left these limits, it is smarter to look for better trends elsewhere.


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Professor Emeritus
School of Public Health
Division of Environmental Health Sciences
University of California
Berkeley, CA 94720-7360
Creative Achievements
Clarified environmental chemistry and identified route of exposure of organophosphate residues poisoning California farm workers. This research provided a scientific foundation for the California regulations protecting farm workers.
Developed methods for the quantification of the effects of uncertainty and variability on environmental health risks which have influenced both risk assessment methods and regulatory strategies.
Led studies of the environmental characteristics underlying the prevalence of the parasitic disease schistosomiasis in the mountainous regions of China and their implications for disease control programs.
Current Research Interests
Mathematical modeling of toxicological and infectious processes
Statistical issues in exposure assessment
Key Publications
R.C. Spear, W.J. Popendorf, J.T. Leffingwell, T.H. Milby, J.E. Davis and W.F. Spencer, Fieldworkers Response to Weathered Residues of Parathion, J. of Occup. Med. 19(6):406–410, 1977.
R.C. Spear, W.J. Popendorf, W.F. Spencer and T.H. Milby, Worker Poisoning Due to Paraoxon Residue, J. of Occup. Med. 19(6):411–414, 1977.
R.C. Spear and G.M. Hornberger, Eutrophication in Peel Inlet: II. Identification of Critical Uncertainties Via Generalized Sensitivity Analysis, Water Research 14:43–49, 1980.
R. C. Spear, E. Seto, S. Liang, M. Birkner, A. Hubbard, D. Qiu, C. Yang, B. Zong, F. Xu, X. Gu, and G. M. Davis, Factors Influencing the Transmission of Schistosoma Japonicum in the Mountains of Sichuan Province, Am. J. Trop. Med. Hyg., 70(10), 48–56, 2004.
S. Wang and R.C. Spear, Exploring the contribution of host susceptibility to epidemiological patterns of S. japonicuminfection using an individually-based model, Am. J. Trop. Med. Hyg., 92, 1245-1252 , 2015 doi10.4269/ajtmh.14-0691.
Research Adviser for Graduate Students
Brief Bio
Dr. Spear is an engineer by training. His research interests focus on the assessment and quantification of human exposures to toxic and infectious agents in the environment. His early work concerned the exposure of agricultural workers to pesticides. In recent years his research has concerned the use of mathematical and statistical techniques in the assessment and control of exposures to both chemical and biological agents. His current work is in collaboration with colleagues both at Berkeley and at the Sichuan Institute of Parasitic Disease in China focused on determinants of the incidence and control of the parasitic disease schistosomiasis. The innovative aspects of this work relate to the integration of traditional epidemiological field data, utilizing geographic information system technology, with that available from both high and low resolution remotely sensed data. Both sources of data are integrated through mathematical models that allow both tracking and forecasting of disease intensity over time. Recent work has focused on the importance of exposure versus individual susceptibility as bases for disease surveillance in low transmission environments.
Curriculum Vitae


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programme planning budgeting monitoring and evaluation - download fast -=DCQ2=-
programme planning budgeting monitoring and evaluation.zip - download 2015
Name: programme planning budgeting monitoring and evaluation.zip
Release: 2015
Type: WinZip
Total Downloads: 14196
License model: Shareware
Language: English
Last updated on: 01/21/2015 04:52:05
Supported OS: Windows 8, Windows 7, Windows Vista, Windows XP
Keys:01/21/2015 04:52:05
Project planning is part of project management, which relates to the use of schedules such as Gantt charts to plan and subsequently report progress within the project ...Use this flexible, self-guided toolkit between USAID training courses to expand and strengthen your design, monitoring and evaluation skills. Work at your own pace …Workshop on Monitoring and Evaluation. The Department of Project Management and Monitoring organized a two day workshop on Monitoring and Evaluation on 20th …Program evaluation is a systematic method for collecting, analyzing, and using information to answer questions about projects, policies and programs, particularly ...3.1 An Overview of Linking Project Design, Annual Planning and M&E. When you "manage for impact", project design, annual planning and M&E become ...... should be used in conducting the OHS program evaluation. ... planning in the safety program ... program monitoring documents ...see also. Kirkpatrick's learning evaluation model; Erik Erikson's Life Stages Theory; Bloom's Taxonomy of learning domains; Free learning and training resources ...Program Evaluation Summary Program Evaluation example The following is an example of a program evaluation, the Executive Summary of the Evaluation of the …iv section 7. integrated monitoring and evaluation plan (imep) ..... 184 section 8: country programme evaluation ...9. Construction Planning 9.1 Basic Concepts in the Development of Construction Plans. Construction planning is a fundamental and challenging activity in the ...
programme planning budgeting monitoring and evaluation
01/21/2015 04:52:05
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