QUEUEING THEORY I think I shall never see a queue as long as this. - Any Customer, Anytime, AnywhereDEFINITION 4.1:A queue is a waiting line of "customers" requiring service fromone or more servers. A queue forms whenever existing demandexceeds the existing capacity of the service facility; that iswhenever arriving customers cannot receive immediate service dueto busy servers.DEFINITION 4.2:Queueing Theory is the study of the waiting line systems. STRUCTURE OF A QUEUEINGMODEL
I. COMPONENTS OF THE QUEUEING PROCESS1.0. The Arrival ProcessA. The calling source may be single or multiple populations.B. The calling source may be finite or infinite.C. Single or bulk arrivals may occur.D. Total, partial or no control of arrivals can be exercised by the queueing system.E. Units can emanate from a deterministic or probabilistic generating process.F. A probabilistic arrival process can be described either by an empirical or a theoretical probability distribution.G. A stationary arrival process may or may not exist.2.0 The Queue Configuration The queue configuration refers to the number of queues in thesystem, their relationship to the servers and spatial consideration.A. A queue may be a single queue or a multiple queue.B. Queues may exist B.1 physically in one place B.2 physically in disparate locations B.3 conceptually B.4 not at allC. A queueing system may impose restriction on the maximum number of units allowed.
3.0 Queue Discipline The following disciplines are possible:A. lf the system is filled to capacity, the arriving unit is rejectedB. Balking - a customer does not join the queueC. Reneging - a customer joins the queue and subsequently decides to leaveD. Collusion - customers collaborate to reduce waiting timeE. Jockeying - a customer switching between multiple queuesF. Cycling - a customer returning to the queue after being given serviceCustomers who do not balk, renege, collude, jockey,cycle, nonrandomly select from among multiple queuesare said to be patient.4.0 Service DisciplineA. First-Come, First-Served (FCFS)B. Last-Come, First-Served (LCFS)C. Service in Random Order (SIRO)D. Round Robin ServiceE. Priority Service . preemptive . non-preemptive
5.0. Service FacilityA. The service facility can have none, one, or multiple servers.B. Multiple servers can be parallel, in series (tandem) or both. Single Queue, Single Server .. .. Multiple Queue, Multiple Servers .. Single Queue, Multiple Servers
Multiple Servers in Series Multiple Servers, both in series and parallel Channels in parallel may be cooperative or uncooperative. By policy, channels can also be variable.C. Service times can be deterministic or probabilistic. Random variables may be specified by an empirical or theoretical distribution.D. State: Dependent state parameters refer to cases where the parameters refer to cases where the parameters are affected by a change of the number of units in the system.E. Breakdowns among servers can also be considered.
CLASSIFICATIONS OF MODELS AND SOLUTIONS1.0. Taxonomy of Queueing Models A model may be represented using the Kendall- LeeNotation: (a/b/c):( d/e/f) where: a = arrival rate distribution b = service rate distribution c = no. of parallel service channels (identical service) d = service discipline e = maximum no. allowed in the system f = calling source Common Notations: M – Poissonl/Exponential rates G - General Distribution of Service Times Ek - Erlangian Distribution2.0 Methods of SolutionA. Analytical: The use of standard queueing models yields analytical results.B. Simulation: Some complex queueing systems cannot be solved analytically. (non-Poisson models)
3.0 Transient vs. Steady StateA. A solution in the transient state is one that is time dependent.B. A solution is in the steady state when it is in statistical equilibrium (time independent)4.0 Analytical Queueing Models - Information Flow In steady state systems, the operating characteristics do not vary with time. Notations λc = effective mean arrival rate λ = λc if queue is infinite λe = λ - [expected number who balk if the queue is finite] W = expected waiting time of a customer in the system Wq = expected waiting time of a customer in the queue L = expected no. of customers in the system Lq = expected number of customers in the queue Po = probability of no customers in the system Pn = probability of n customers in the system ρ = traffic intensity= λ/μ ρc = effective traffic intensity= λe/μ
GENERAL RELATIONSHIPS: (LITTLES FORMULA)The following expressions are valid for all queueing models.Theserelationships were developed by J. Little L=λW Lq = λWq W =Wq+1/μ L = L q + λ/ μNote: lf the queue is finite, λ is replaced by λe EXPONENTIAL QUEUEING MODELSIn the models that will be presented the following assumptions holdtrue for any model:1. The customers of the queueing system are patient customers.2. The service discipline is general discipline (GD), which means that the derivations do not consider any specific type of service discipline.The derivation of the queueing models involve the use of a set ofdifference-differential equations which allow the determination ofthe state probabilities. These state probabilities can also becalculated by the use of the following principle:Rate-Equality Principle: The rate at which the process enters state nequals the rate at which it leaves state n.
CASE 1: SINGLE CHANNEL-POISSON/EXPONENTIALMODEL [(M/M/1):(GD/ α /α)]Characteristics:1. Input population is infinite.2. Arrival rate has a Poisson Distribution3. There is only one server.4. Service time is exponentially distributed with mean 1/μ. [λ<μ]5. System capacity is infinite. .6. Balking and reneging are not allowed.Using the rate-equality principle, we obtain our first equation for thistype of system: λPo=μP1To understand the above relationship, consider state 0. When instate 0, the process can leave this state only by an arrival. Since thearrival rate is λ and the proportion of the time that the process is instate 0 is given by Po, it follows that the rate at which the processleaves state 0 is λPo. On the other hand, state 0 can only be reachedfrom state 1 via a departure. That is, if there is a single customer inthe system and he completes service, then the system becomesempty. Since the service rate is μ and the proportion of the time thatthe system has exactly once customer is P1, it follows that the rateat which the process enters 0 is μPl. The balance equations usingthis principle for any n can now be written as: State Rate at which the process leaves = rate at which it enters0 λ P0=μP1.
n ≥1 (λ + μ)Pn = λP + μP n-1 n+lIn order to solve the above equations, we rewrite them toobtainSolving in terms of P0yields:In order to determine P0, we use the fact that the Pn mustsum to 1, and thus or
Note that for the above equations to be valid, it is necessary for to be less than 1 so that the sun of the geometric progression customers in the system at any time, we useThe last equation follows upon application of the algebraic identityThe rest of the steady state queueing statistics can be calculatedusing the expression for L and Littles Formula. A summary ofthe queueing formulas for Case I is given below.
SUMMARY OF CASE 1 FORMULASCASE II : MULTIPLE SERVER, POISSON/
EXPONENTIAL MODEL [(M/M/C):(GD/∞/∞ )] The assumptions of Case II are the same as Case 1 except thatthe number of service channels is more than one. For this case, theservice rate of the system is given by:cµ η≥ cηµ η<cThus, a multiple server model is equivalent to a single-serversystem with service rate varying with η. λη = λ & µη = ηµ η < cUsing the equality rate principle we have the following balanceequations: =0
But:Therefore:η≥cIf ρ=λ/µTo solve for , we note that . Hence:
CASE III: SINGLE CHANNEL.POISSON ARRIVALS, ARBITRARY SERVICE TIME: Pollaczek - Khintchine Formula [(M/G/l): (GD/∞/∞)]This case is similar to Case 1 except that the service ratedistribution is arbitrary.Let: N = no. of units in the queueing system immediately after a unit departs T = the time needed to service the unit that follows the one departing (unit 1) at the beginning of the time count. K= no. of new arrivals units the system during the time needed to service the unit that follows the one departing (unit 1) Nl = no. of units left in the system when the unit (1) departsThen: Nl = N + K – 1; if N = 0 =KLet: a = 1 if N = 0 a = 0 if N > 0 a*N =0Then: Nl = N + K + a – 1In a steady state system: ~.E(N)= E( )E( )= E[ ]E( )=
E (a) = -E (K) + 1 = =Since a = 0 or 1: a*N= 0 ButTherefore :but E(a) = 1 - E(K)Then:
If the arrival rate is Poisson , E(K/t) = λtBut in a Poisson Distribution: Mean = Variance E (K2/t) = λt + (λt)2
Substituting and solving for E(N):The other quantities can be solved using the general relationshipsderived by Little.
CASE IV: POISSON ARRIVAL AND SERVICE RATE,INFINITE NUMBER OF SERVERS: Self-Service Model[(M/M/∞): (GD/∞/∞)]Consider a multiple server system. The equivalent single serversystem if the number of servers is infinite would be:From the multiple server system:But
Therefore:Since the number of servers is infinite: Lq = Wq = 0.Solving for L:Again, W could be solved using Little’s Formula.
CASE V: SINGLE CHANNEL, POISSON/EXPONENTIAL MODEL, FINITE QUEUE [(M/M/1) : (GD/m/∞)]This case is similar to Case 1 except that the queue is finite, i.e.,when the total number of customers in the system reaches theallowable limit, all arrivals balk.Let m = maximum number allowed in systemThe balance equations are obtained in the same manner as before.n=0:n=1n=m-1:n=m:But
Where is a finite geometric series with sumTherefore:Now:...
CASE VI : MULTIPLE CHANNEL, POISSONEXPONENTIAL MODEL, FINITE QUEUE [(M/M/c):(GD/m/∞)]This case is an extension of Case V. We assume that the number ofservice channels is more than one. For this system:The balance equations are similar to Case II. The manipulation ofequations is basically the same. The following are the results.
As in Case V, we solve for:This expression is used in solving for the other statistics.
CASE VII : MACHINE SERVICING MODEL [(M/M/R): (GD/K/K)]This model assumes that R repairmen are available for servicing atotal of K machines. Since a broken machine cannot generate newcalls while in service, this model is an example of finite callingsource. This model can be treated as a special case of the singleserver, infinite queue model. Moreover, the arrival rate λ isdefined as the rate of breakdown per machine. Therefore:The balance equations yield the following formulas for the steadystate system:
The other measures are given by:To solve for the effective arrival rate, we determine:
ECONOMIC CONDITIONSCOSTS INVOLVED IN THE QUEUEING SYSTEM1. FACILITY COST - cost of (acquiring) services facilities Construction (capital investment) expressed by interest and amortization Cost of operation: labor, energy & materials Cost of maintenance & repair Other Costs such as insurance, taxes, rental of space2. WAITING COST - may include ill-will due to poor service, opportunity loss of customers who get impatient and leave or a possible loss of repeat business due to dissatisfaction.The total cost of the queueing system is given by: TC = SC+WC where: SC = facility (service cost)cost WC = waiting cost or the cost of waiting (in queue & while being served) per unit time TC = Total CostLet Cw = cost of having 1 customer wait per unit time
Then WCw = average waiting cost per customerBut since λ customers arrive per unit time:WC = λ WCw = LCwThe behavior of the different cost Component is depicted in thefollowing graph:Management Objective: Cost Minimization or Achieving a Desired Service LevelAn example of a desired service level is the reduction of waitingtime of customers. The minimization of cost would involve theminimization of the sum of service cost and waiting cost.The decision is a matter of organizational policy and influenced bycompetition and consumer pressure.