The inter-departure time distribution and lag-1 joint moments

6.4 Analysis of the departure process

6.4.4 The inter-departure time distribution and lag-1 joint moments

In this section we determine an expression for the moments of the inter-departure time dis-tribution as well as the joint lag-1moments via Theorem23. Based on the lag-1moments it is possible to plug the MMAP[K]/PH[K]/1 FCFS queue into the queueing network analysis framework introduced in Chapter8.

We start by defining(v⁽₁^k⁾)_j, for j ∈ {1, . . . ,N_A}andk ∈ {1, . . . ,K}, as the probability that an arbitrary departing customer leaves a single customer behind, the type of which isk, while theMMAPphase at the departure epoch is j. Denotev⁽₁^k⁾ as the vector with entry j equal to(v⁽₁^k⁾)_j.

Lemma 6. The1×N_Avectorsv⁽₁^k⁾can be computed asv⁽₁^k⁾= ρa(0)Y₁⁽^k⁾/λ, where the matri-cesY₁⁽^k⁾, fork =_{1, . . . ,}K, are the unique solutions to the Sylvester matrix equations

TY⁽^k⁾+Y⁽^k⁾D = −Y D . (209)

6.4 analysis of the departure process 93 Proof. A departure leaves a single (typek) customer behind if theMMAPgenerates a single (type k) arrival during the sojourn time of the departing customer. By conditioning on the arrival time of this typekcustomer we get

v⁽₁^k⁾ =

Z _∞

x=0a_D(x)

Z _x

a=0e^D⁰^aD_ke^D⁰⁽^x⁻^a⁾da dx, (210) which yields (due to Theorem37)

v⁽₁^k⁾ = ^ρa(0) X⁽^k⁾is the unique solution of the Sylvester matrix equation

T X⁽^k⁾+X⁽^k⁾

Theorem 24. TheLST f_H^∗(s)of the inter-departure time distribution is given by f_H^∗(s) =^h(α−v₀)(−D₀)⁻¹+v₀(sI−D₀)⁻¹ⁱ customer isk, can be expressed as

f_H⁽^k^)∗(s₁,s2) =^h(α−v₀)(−D₀)⁻¹+v₀(s₁I−D₀)⁻¹ⁱD_k· also be proven algebraically as follows. Combining (205) and (204) yields

v^∗_k,1(s) = ^ρa(0)

94 analysis of the mmap[k]/ph[k]/1-fcfs qeue

Equation (197) and Lemma5imply that the first term reduces to v₀(sI−D₀)⁻¹D_kM_k,1^∗ (s), Instead of computing the moments directly, we introduce the so-called reduced moments

Eˆ(Hⁿ) =E(Hⁿ)/n!, Eˆ(S_kⁿ) =E(S_kⁿ)/n!, because they make the forthcoming expressions simpler.

Corollary 3. Thenth reduced moment of the inter-departure time distribution is given by Eˆ(Hⁿ) = which establishes the result as

α(−D₀)⁻¹D_k1=θ inter-departure times. Again, for simplicity we use the reduced moments instead of the standard ones. The(n₁,n₂)th reduced joint moment is denoted byηˆ_n⁽^k₁⁾_,n₂ and is obtained from theLST

6.4 analysis of the departure process 95

Corollary 4. The(n₁,n2)th reduced joint moment of the inter-departure times are given by ˆ

η⁽_n^k₁⁾_,n₂ =

αP_kEˆ(S_kⁿ¹) +v₀

n₁

∑

(−D₀)^−`P_kEˆ(S_kⁿ¹^−`)

# K q

∑

P_q1Eˆ(Sⁿ_q²)

v⁽₁^k⁾M¯_k,1ⁿ¹ +v₀

∑

(−D₀)^−`P_kM¯_k,1ⁿ¹^−`

# n2

∑

(−D₀)⁻^d

∑

K q=1

P_q1Eˆ(S_qⁿ²⁻^d)

! ,

(214) whereM¯_k,1ⁿ is defined and computed as follows:

M¯_k,1ⁿ = (−1)ⁿ n!

dⁿ

dsⁿM_k,1^∗ (s)|_s₌₀= (σ_k⊗I)((−S_k)⊕D₀)⁻ⁿ⁻¹(−S_k1⊗I). (215)

7

A N A LY S I S O F T H E M M A P [ K ] / P H [ K ] / 1 P R I O R I T Y Q U E U E

In the MMAP[K]/PH[K]/1 queueKtypes (classes) of customers are distinguished. The arrival process of customers is described by aMMAP, and the service times arePHdistributed. There is a single server, which always picks the customer having the highest priority for service.

If the ongoing service can not be interrupted when a higher priority customer arrives, the service is called to benon-preemptive. In thepreemptive resumecase (also referred to as the preemptive case for simplicity), however, the service of customers can be interrupted, and resumed later when all higher priority customers leave the system.

To introduce the analysis approach, the two-class case (K = 2) is considered throughout the chapter, and the extension to the general case (K>2) is provided in [49].

Similar to Chapter6, theMMAPcharacterizing the arrivals is given by the size N_A×N_A matricesD₀,DH andDL containing the rates of internal transitions and transitions accom-panied by high and low priority customers. The mean arrival rate of high (low) priority cus-tomers is denoted byλ_H(λL), and it is calculated byλ_H =θD_H1(λL=θD_L1), respectively, whereθis the stationary distribution of the phase process of theMMAP(see Section3.1.2).

The random variable representing the service times of the low priority customers S_L is PHdistributed withN_Lphases, characterized byσ_L,S_L ands_L. Row vectorσ_Lis the initial vector, matrixSL is the transient generator and column vectors_Lholds the transition rates to the absorbing state, thuss_L = −S_L1. The mean service rate isµ_L = 1/E(S_L). The PH distribution corresponding to the high priority service times and its properties are defined similarly, using subscriptHinstead ofL.

The load of the queue isρ =λ_H/µ_H+λ_L/µ_L. In this chapterρ <1is assumed.

7.1 analysis of the preemptive resume priority qeue

Priority queues are extensively studied since the middle of the last century [61], starting with the most basic variant with Poisson arrival process and exponentially distributed service times.

In the last two decades most research activity on priority queues has considered more general arrival processes likeMAPsorMMAPs.

In [78] the MAP/G/1 preemptive priority queue is analyzed based on the workload process, and theLSTof the sojourn time distribution of the customers is derived. The non-preemptive case is investigated in [79] and [80], where theLSTof the sojourn time, the moments of the sojourn time, theGFof the queue length, the queue length moments and the queue length probabilities are provided.

After this overview one may think that not too much has left to be done in the field ofMAP driven priority queues. However, all the aforementioned results assume a general distribution for the service times, which makes the solution complex and often difficult to implement in a proper way (in the numerical sense). To address this issue the generally distributed service

98 analysis of the mmap[k]/ph[k]/1 priority qeue

t V(t) Workload after low priority arrivals

Inter-arrival time

Service time

Figure 29.: The workload process of the priority queue

times can be replaced byPHdistributed ones in the hope of the simpler and numerically more tractable solution.

In [3] the (discrete-time) MAP/PH/1 priority queue is considered by representing the state space with aQBDand exploiting the special structure of the related fundamental matrices.

While this approach is elegant and seems promising, there are some computational bottle-necks (as pointed out in [48]). There have been efforts to make it more efficient (see [45] and [48]), but apart from the queue length moments all performance measures can be computed only in case of a very limited number of phases.

Our approach is based on the analysis of the workload process, just like [78] in the context of MAP/G/1 preemptive priority queues. However, by exploiting the technical simplicity of thePHdistributed service times we are able to arrive to a more intuitive, simpler to implement and numerically more beneficial solution.

In document E F F I C I E N T M AT R I X-A N A LY T I C S O L U T I O N O F M U L T I -T Y P E Q U E U E I N G S Y S T E M S W I T H C O R R E L AT E D T R A F F I C gábor horváth Dissertation submitted to the Hungarian Academy of Sciences for the degree of Doctor of (Pldal 104-110)