APPENDIX A SYMBOLS AND ABBREVIATIONS This appendix contains definitions of common symbols and abbreviations used frequently and consistently throughout the text. Symbols that are used only occasionally in isolated sections of the text are not always included here. The symbols are listed in alphabetical order. Greek symbols are filed according to their English names; for example, A (lambda) is found under L. Within an alphabetical category, Latin symbols precede Greek. Listed at the end are nonliteral symbols such as primes and asterisks. Fundamentals of Queueing Theory, Fourth Edition. By D. Gross, 1. F. ShortIe, 1. M. Thompson, and C. M. Harris Copyright 2008 John Wiley & Sons, Inc. 439
440 SYMBOLS AND ABBREVIATIONS A/B/X/Y/Z a.s. A(t) a(t) B(t) b(t) b n C C CDF CK CTMC CV C s Cn C w C(t) C(z) c(t) D d DFR DTMC df /j.yn Ek E[ ] G GCD Notation for describing queueing models, where A indicates interarrival pattern, B indicates service pattern, X indicates number of channels, Y indicates system capacity limit, Z indicates queue discipline Almost surely Cumulative distribution function of interarrival times Probability density of interarrival times Cumulative distribution function of service times Probability density of service times (1) Probability of n services during an interarrival time; (2) discouragement function in queueing models with balking (l) Arbitrary constant; (2) cost per unit time, often used in functional notation as C (. ); (3) coefficient of variation (= CT / fl ) Number of parallel channels (servers) Cumulative distribution function Chapman - Kolmogorov Continuous-time Markov chain Coefficient of variation (= CT / fl) Marginal cost of a server per unit time Probability that the batch size is n Cost of customer wait per unit time CDF of the interdeparture process Probability generating function of {c n } Probability density of the interdeparture process (1) Deterministic interarrival or service times; (2) linear difference operator, DX n = X n + 1 ; (3) linear differential operator, Dy(x) = dy/dx Mean observed interdeparture time of a queueing system Decreasing failure rate Discrete-time Markov chain Distribution function First finite difference; that is, /j.yn = Yn+l - Yn Erlang type-k distributed interarrival or service times Expected value Mean time spent in state i before going to j First-come, first-served queue discipline Joint probability that n are in the system at time t after the last departure and t is less than the interdeparture time Conditional probability that, given that a process begins in state i and next goes to state j, the transition time is :::; t (a conditional CDF) Probability that state j of a process is ever reached from state i Probability that the first passage of a process from state i to state j occurs in exactly n steps General distribution for service and/or interarrival times Greatest common divisor
SYMBOLS AND ABBREVIATIONS 441 GD G(N) G(t) G(z) H H(z,y) h(u) I IFR lid Iu InC) i(t) in(t) K(z) Ki(Z) k n kn,i L LCFS LST LT L(D) General queue discipline Normalizing constant in a closed network Cumulative distribution function of the busy period for M / G /1 and G / 1\-1 /1 models Generating function associated with Erlang-service steady-state probabilities {Pn,d Conditional probability that, given that a process starts in state i, the time to the next transition is :::; t (a conditional CDF) External flow rate to node i of a network Mixture of k exponentials used as distribution for interarrival and/or service times Hyperexponential (a balanced H 2 ) distribution for service and/or interarrival times (1) Probability generating function for {Pn,d; (2) joint generating function for a two-priority queueing model Cumulative distribution function of time until first transition of a process into state j beginning at state i Generating function associated with Pmr(z) for a two-priority queueing model Failure or hazard rate of a probability distribution Phase of service the customer is in for Erlang service models (a random Increasing failure rate Independent and identically distributed Expected useful server idle time Modified Bessel function of the first kind Probability of a server being idle for a time> t (a complementary CDF) Conditional probability that one of the c - n idle servers remains idle for a time> t (a conditional complementary CDF) Probability density of idle time Regular Bessel function System capacity limit (truncation point of system size) Greatest queue length at which an arrival would balk (a random Probability generating function of {k n } Probability generating function of {kn,d Probability of n arrivals during a service time Probability of n arrivals during a service time, given i in the system when service began Expected system size Last-come, first-served queue discipline Laplace-Stielties transform Laplace transform Expected system size at departure points
442 SYMBOLS AND ABBREVIATIONS L(D) q(k) L( ),C (-) A A An M MC MGF MLE MOM Jvlx(t) rni mij mjj J1 J1(B) J1n o(~t) w P PDE PK Expected number of phases in the system of an Erlang queueing model (1) Expected number of customers of type n in system; (2) expected system size at station n in a series or cyclic queue The kth factorial moment of system size Expected queue size Expected queue size of nonempty queues Expected queue size at departure points Expected number of phases in the queue of an Erlang queueing model (1) Expected queue size for customers of type n; (2) expected queue size in front of station n in a series or cyclic queue The kth factorial moment of the departure-point queue size Likelihood function (1) Log-likelihood function; (2) Laplace transform Minimum diagonal element of Q Mean arrival rate (independent of system size) (1) Mean arrival rate when there are n in the system; (2) mean arrival rate of customers of type n (1) Poisson arrival or service process (or equivalently exponential interarrival or service times); (2) finite population size Markov chain Moment generating function Maximum-likelihood estimator Method of moments (estimator) Moment generating function of the random variable X Mean time a process spends in state i during a visit Mean first passage time of a process from state i to state j Mean recurrence time of a process to state j Mean service rate (independent of system size) Mean service rate for a bulk queueing model (1) Mean service rate when there are n in the system; (2) mean service rate of server n; (3) mean service rate for customers of type n Steady-state number in the system (a random Steady-state number in the queue (a random Number in the system at time t (a random Number in the queue at time t (a random Number of observed arrivals to a system, to an empty system, and to a busy system, respectively (na = nae + nb) Order ~t; that is, limtl,t-to u( ~t) / ~t = 0 Expected remaining work Single-step transition probability matrix of a DTMC Partial differential equation Pollaczek - Khintchine formula
SYMBOLS AND ABBREVIATIONS 443 PR P(z), P(z, t) Pmr(z) p Pn (B) pn (P) pn Pn(t) Pij Pn,i (n) Pij Pn,i(t) Pmnr(t) Pi,j (U, s) Pnl,n2,...,nk (t) Pnl,n2,...,nk p-c II(z) 1T qn R R(t) Re RK RSS RV r Priority queue discipline Probability generating function of {Pn} and {Pn ( t) }, respectively Probability generating function of priority steady-state probabilities {P mnr } Steady-state probability vector of a CTMC (1) Steady-state probability of n in the system; (2) steady-state probability that a CTMC is in state n Steady-state probability of n in a bulk queueing system Steady-state probability of n phases in an Erlang queueing system Probability of n in the system at time t Single-step transition probability of going from state i to state j Steady-state probabilities for Erlang models of n in the system and the customer in service (if service is Erlangian) or next to arrive (if arrivals are Erlangian) in phase i Transition probability of going from state i to state j in n steps Probability that, in an Erlang queueing model at time t, n are in the system and the customer in service (if the service is Erlangian) or next to arrive (if arrivals are Erlangian) is in phase i Probability at time t of m, units of priority 1, n units of priority 2 in the system, and a unit of priority r in service (r = 1 or 2) Transition probability of moving from state i to state j in time beginning at u and ending at s Probability of nl customers at station 1, n2 at station 2,...,nk at station k in a series queue at time t Steady-state probability of Pnl,n2,...,nk (t) Predictor - corrector Probability generating function of {'7T n } Steady-state probability vector of a DTMC (1) Steady-state probability of n in the system at a departure point; (2) steady-state probability that a DTMC is in state n Infinitesimal generator matrix of a CTMC Joint conditional probability that, given that a process begins in state i, the next transition will be to state j in an amount of time t (a conditional CDF) Steady-state probability that an arriving customer finds n in the system Network routing probability matrix Distribution function of remaining service time Real portion of a complex number Runge-Kutta Random selection for service Random variable Defined as )..1 JL for multichannel models; defined as )..1 kjl for Erlang service models
444 SYMBOLS AND ABBREVIATIONS r(n) p S SMP s(n) S(n) k Sk(SU So T T(n) TA Ti Ts T busy Tq Tb,i tb, te, t U(t) The ith root of a polynominal equation (if there is only one root, TO is used) The nth uniform (0, 1) random number Routing probability in a queueing network of a customer going to station j after being served at station i Reneging function Traffic intensity (=)../ f-l for single-channel and all network models, and =)../ CJl for other multichannel models) Steady-state service time (a random Semi -Markov process Service time of the nth arriving customer (a random Service time of the nth arriving customer of type k Time it takes to serve nk(nu waiting customers of type k (a random Time required to finish customer in service (remaining time of service; a random Probability that n servers are busy (c - n idle) in a multichannel system Sample standard deviation of the random variable X Variance of the service-time distribution Variance of the interarrival-time distribution Sum of traffic intensities in a priority queueing model; that is, (}k = ~)..d Jli (1) Time spent in system (a random, with expected value W; (2) steady-state interarrival time (a random ; (3) steady-state interdeparture time (a random Interarrival time between the nth and (n + l)st customers (a random Instant of arrival Length of time a stochastic process spends in state i (a random Instant of service completion Length of a busy period (a random Time spent in queue (a random, with expected value Wq Length of i channel busy period for M / M / C (a random Observed time a system is busy, observed time a system is empty, and total observed time, respectively (t = tb + te) Time at which the ith arrival to a Poisson process occurred Steady-state difference between service time and interarrival time, U = S - T (a random Service time of nth customer minus interarrival time between customer n + 1 and n; that is u(n) = s(n) - T(n) (a random (1) Cumulative distribution function of U = S - T; (2) cumulative distribution function of the time back to the most recent transition
SYMBOLS AND ABBREVIATIONS 445 Var[ ] V V(t) W w(n) W k Wq Wq,k W(t) w:(h) q w:(n) q X(t) [x] E * (nc) Cumulative distribution function of u(n) Cumulative distribution function of the time back to the most recent transition, given the process starts in state i Variance Expected virtual wait Virtual waiting-time function (1) Steady-state probability of a semi-markov process being in state j; (2) relative throughput in a closed network Expected waiting time in system Waiting time including service at station n of a series or cyclic queue Ordinary kth moment of waiting time in system Expected waiting time in queue Regular kth moment of waiting time in queue Cumulative distribution function of waiting time in system Expected time in queue for a system in heavy traffic (1) Waiting time in queue for the nth arriving customer (a random ; (2) expected wait in queue for customers of priority class n; (3) expected time in queue at station n of a network of queues Cumulative distribution function of waiting time in queue Probability in M / M / c model that the delay undergone by an arbitrary arrival who joined when c + j were in the system is more than t Stochastic process with state space X and parameter t Observed interval of a queueing system, and observed interval of type i (busy, empty, etc.) of a queueing system, respectively Greatest integer value ~ x Approximately equal to Asymptotic to Set membership (1) LST; (2) used for various other purposes as specifically defined in text Laplace transform Binomial coefficient, n! / [( n - c)!c!] [.] Batch queueing model (. ) (1) Order of convolution; (2) order of differentiation, (3) number of steps (transitions) in a discrete-parameter Markov chain. (1) Differentiation; (2) conditional; for example, p~ is a conditional probability distribution of n in the system given system not empty; (3) used for various other purposes as specifically defined in text (1) Complementary CDF; (2) used for various other purposes as specifically defined in text