Discrete Stochastic Delivery Modeling - Kartuzy, PL 83-300

A Non-Homogeneous Markov and Bayesian Recovery Model for Right-Censored Discrete Transit in Polish Postal Infrastructure

Abstract

We construct a non-homogeneous discrete-time Markov chain over the Polish postal pipeline from dispatch to delivery at Kartuzy, postal code 83-300, gated by a business-day indicator. Total transit latency is modeled as the convolution of an internal dispatch latency and a postal transit latency via probability generating functions, with parameters calibrated to a single empirical anchor (Mom baseline, Aug 14 - Aug 22). A conjugate Bayesian update yields a posterior transit distribution, and renewal-theoretic conditioning on right-censoring at operational day 4 (Friday, Aug 21) produces a truncated arrival distribution. We derive the posterior mass function, its expectation, variance and skewness, and a 97% credible interval for arrival at Kartuzy.

Axiomatic probability space and filtration

Let time be discrete, $t \in N_{0}$ . The sample space carries the arrival calendar day together with the latent latencies that determine it.

Ω = {ω = (T_{arr}, D_{disp}, D_{post})}, F = 2^{Ω}, Pr : F \to [0, 1]

(1)

The triple $(Ω, F, Pr)$ is a probability space. The filtration $(F_{t})_{t \geq 0}$ is the information revealed up to day $t$ : the dispatch status, the observed non-arrival, and the business-day calendar. It is a filtration because $F_{s} \subseteq F_{t}$ for $s \leq t$ , and the arrival time is a stopping time.

T_{arr} is a stopping time: {T_{arr} \leq t} \in F_{t} \forall t \geq 0

(2)

Non-homogeneous discrete-time Markov chain

The physical pipeline is a Markov chain on the state vector $S_{t} \in {S_{0}, \dots, S_{5}}$ :

S_{0} = Dispatch, S_{1} = Personalization, S_{2} = Warsaw Sorting, S_{3} = Gdansk Regional Hub, S_{4} = Kartuzy Local Hub, S_{5} = Delivery

(3)

Let $B (t) \in {0, 1}$ be the business-day gating indicator and $Q$ the nominal one-step transition matrix. The time-varying transition matrix is

P (t) = I + B (t) (Q - I)

(4)

On a business day $B (t) = 1$ and $P (t) = Q$ ; on a non-business day $B (t) = 0$ and $P (t) = I$ , so the chain holds in place. With advance probabilities $p_{i}$ from state $S_{i}$ :

Q = 1 - p_{0} 00000 p_{0} 1 - p_{1} 0000 0 p_{1} 1 - p_{2} 000 00 p_{2} 1 - p_{3} 00 000 p_{3} 1 - p_{4} 0 0000 p_{4} 1

(5)

Lemma - Stochasticity and absorption

Each row of $Q$ sums to 1, so $Q$ is row-stochastic. State $S_{5}$ is absorbing, $Pr (S_{t + 1} = S_{5} ∣ S_{t} = S_{5}) = 1$ , and the arrival time is the first-passage time to $S_{5}$ .

Convolution of latency via generating functions

Total operational latency is the sum of independent components. The internal dispatch latency is two-point,

D_{disp} \in {1, 2}, Pr (D_{disp} = 1) = Pr (D_{disp} = 2) = 0.50

(6)

and the postal transit latency is calibrated on the Mom baseline $t_{0, M} = Aug 14$ , $t_{arr, M} = Aug 22$ , a six-business-day interval:

D_{post} \in {3, 4, 5, 6}, Pr (D_{post} = 3) = 0.25, Pr (D_{post} = 4) = 0.50, Pr (D_{post} = 5) = 0.20, Pr (D_{post} = 6) = 0.05

(7)

The probability generating function of the total latency factors:

G_{L} (z) = E [z^{L}] = G_{D_{disp}} (z) G_{D_{post}} (z)

(8)

G_{D_{disp}} (z) = 0.50 z + 0.50 z^{2}, G_{D_{post}} (z) = 0.25 z^{3} + 0.50 z^{4} + 0.20 z^{5} + 0.05 z^{6}

(9)

Expanding the product is the discrete convolution, with support $L \in {4, 5, 6, 7, 8}$ :

Pr (L = l) = j \sum Pr (D_{disp} = j) Pr (D_{post} = l - j)

(10)

Theorem - Moments from the generating function

E [L] = G_{L}^{'} (1), E [L (L - 1)] = G_{L}^{''} (1), Var (L) = G_{L}^{''} (1) + G_{L}^{'} (1) - (G_{L}^{'} (1))^{2}

Proof. Since $G_{L} (z) = \sum_{l} Pr (L = l) z^{l}$ , termwise differentiation at $z = 1$ yields the factorial moments, and the variance identity follows from $Var (L) = E [L^{2}] - E [L]^{2}$ .

The unconditioned mass function over $L \in {4, 5, 6, 7, 8}$ :

l	Pr(L = l)
4	0.125
5	0.375
6	0.350
7	0.125
8	0.025

Bayesian likelihood and conjugate update

Model the postal transit probabilities as $θ = (θ_{3}, θ_{4}, θ_{5}, θ_{6})$ over $D_{post} \in {3, 4, 5, 6}$ . The single empirical anchor is the Mom baseline, a six-business-day total latency over $Aug 14 \to Aug 22$ .

Assumptions. Only one empirical observation is available, so the transit distribution is weakly identified. We adopt a symmetric Dirichlet prior and treat the supplied transit distribution as the posterior calibrated to this single anchor. No additional empirical observations are asserted.

π (θ) = Dirichlet (θ ∣ α), α = (α_{3}, α_{4}, α_{5}, α_{6})

(11)

L (θ ∣ y_{mom}) = k \prod θ_{k}^{y_{mom, k}}

(12)

The Dirichlet-multinomial pair is conjugate, so the posterior is again Dirichlet:

p (θ ∣ y_{mom}) \propto L (θ ∣ y_{mom}) π (θ) = Dirichlet (θ ∣ α + y_{mom})

(13)

The posterior mode is $θ^_{k} = (α_{k} + y_{mom, k} - 1) / (\sum_{k} (α_{k} + y_{mom, k}) - K)$ with $K = 4$ . Calibrating to the supplied transit distribution gives $θ^= (0.25, 0.50, 0.20, 0.05)$ .

Renewal theory and discrete hazard rate

The arrival time is a first-passage (renewal) time. Define survival and hazard:

F (t) = Pr (T_{arr} \leq t), S (t) = 1 - F (t) = Pr (T_{arr} > t)

(14)

λ (t) = Pr (T_{arr} = t ∣ T_{arr} \geq t) = \frac{f ( t )}{1 - F ( t - 1 )}

(15)

The renewal identities hold:

f (t) = S (t - 1) - S (t), S (t) = k \leq t \prod (1 - λ (k)), E [T_{arr}] = t \geq 0 \sum S (t)

(16)

Evaluated on the conditioned distribution (Section 6):

λ (5) = 0.4286, λ (6) = 0.70, λ (7) = 0.8333, λ (8) = 1.00

(17)

Right-censoring at t=4 and conditional distribution

Arrival at operational day $t = 4$ (Friday, Aug 21) was not observed. It is treated as right-censored and rejected from the support, conditioning on $T_{arr} \geq Aug 24$ .

Theorem - Truncated conditional distribution

Pr (T_{arr} = t ∣ T_{arr} \geq Aug 24) = \frac{Pr ( L = t )}{Pr ( L \geq 5 )}, t \in {5, 6, 7, 8}

Proof. Since ${T_{arr} = t} \subseteq {T_{arr} \geq 5}$ for $t \geq 5$ , the conditional probability equals $Pr (L = t) / Pr (L \geq 5)$ . The normalizing mass is $Pr (L \geq 5) = 0.875$ , and normalization is exact.

The resulting posterior mass function over the support ${5, 6, 7, 8}$ :

Pr (T_{arr} = Aug 24 ∣ T_{arr} \geq Aug 24) = 0.4286, Pr (T_{arr} = Aug 25 ∣ \cdot) = 0.4000

(18)

Pr (T_{arr} = Aug 26 ∣ \cdot) = 0.1429, Pr (T_{arr} = Aug 27 ∣ \cdot) = 0.0286

(19)

The support and normalization are summarized in the calendar table:

t	Date	Day	Note
0	Aug 17	Mon	dispatch
1	Aug 18	Tue
2	Aug 19	Wed
3	Aug 20	Thu
4	Aug 21	Fri	right-censored
5	Aug 24	Mon
6	Aug 25	Tue
7	Aug 26	Wed
8	Aug 27	Thu

Moments, sensitivity and credible interval

Using the exact conditioned probabilities $(\frac{3}{7}, \frac{2}{5}, \frac{1}{7}, \frac{1}{35})$ over $t \in {5, 6, 7, 8}$ :

Theorem - Expectation

E [T_{arr}] = t = 5 \sum 8 t Pr (T_{arr} = t) = \frac{202}{35} = 5.7714

Proof. $5 \cdot \frac{3}{7} + 6 \cdot \frac{2}{5} + 7 \cdot \frac{1}{7} + 8 \cdot \frac{1}{35} = \frac{202}{35}$ . The expected operational day rounds to $t = 6$ , Tuesday, Aug 25.

Theorem - Variance

Var (T_{arr}) = E [T_{arr}^{2}] - E [T_{arr}]^{2} = \frac{776}{1225} = 0.6335, σ = 0.7959

Proof. $E [T_{arr}^{2}] = \frac{1188}{35}$ and $\frac{1188}{35} - (\frac{202}{35})^{2} = \frac{776}{1225}$ .

Theorem - Skewness

Skew (T_{arr}) = \frac{E [( T _{arr} - μ ) ^{3} ]}{σ ^{3}} = 0.772

Proof. With third raw moment $E [T_{arr}^{3}] = \frac{7126}{35}$ and $E [(T - μ)^{3}] = E [T^{3}] - 3 μ E [T^{2}] + 2 μ^{3}$ , the positive skew indicates a right tail toward Aug 27.

The posterior mass function, summing to 100% subject to rounding:

Date	Day	Pr	Distribution
Aug 24	Mon	42.86%	0.429
Aug 25	Tue	40.00%	0.400
Aug 26	Wed	14.29%	0.143
Aug 27	Thu	2.85%	0.029

Parameter sensitivity table - illustrative diagnostic only. The values below vary the dispatch-latency split $p = Pr (D_{disp} = 1)$ while holding the postal transit distribution fixed. They are a numerical check of the model's own structure, not new empirical observations.

p	E[L]	E[T_arr \| censored]
0.40	5.6500	5.8333
0.45	5.6000	5.8028
0.50	5.5500	5.7714
0.55	5.5000	5.7391
0.60	5.4500	5.7059

The baseline $p = 0.50$ reproduces $E [T_{arr}] = 5.7714$ , i.e. Tuesday, Aug 25.

Theorem - 97% credible interval

Pr (Aug 24 \leq T_{arr} \leq Aug 26) = \frac{3}{7} + \frac{2}{5} + \frac{1}{7} = \frac{34}{35} = 0.9714 \approx 0.97

CI_{0.97} = [Aug 24, Aug 26]

The model assigns 97% of the conditioned mass to Aug 24 - Aug 26, centered on the expected arrival of Tuesday, Aug 25.